Okay, I checked the save, so I have some answers to the questions: [Good lord, this turned out much longer than I expected. I ended up spending an hour writing this! Hope it is of some use.]
So... on to my questions
What am I doing wrong?
Are my defenses set incorrectly?
What could I do better?
Is there something I'm missing?
First, you are playing Fallen Spire on 6/10 rather than the default 4/10, so it is to be expected that things are tougher for you than on 4/10, and you are missing many obvious improvements to your approach. I can fully understand that you are experiencing pain in the given situation.
Second, a small but obvious thing - you are placing your gravity turrets poorly, directly along the line of advance of enemy ships. You should put them along lines parallel to the line of advance such that they merely overlap it (which means that short ranged enemy ships won't even be able to hit the turret without detouring from the advance) and, since you've got Zenith Traders, if you slap down a Radar Jammer 2 as well even most long-range enemy ships will be unable to hit the turrets without departing from the line. Which they eventually will, of course, but time the enemy wastes is time you can take advantage of - and which your defences surely will.
Third, you have conquered... unwisely. Lord knows that your map isn't trivial to make sense of, but you seem to have been going for a maximum of exposure to hostile threats with minimal safe interior space and minimal use of defensive depth. Nearly everything is either exposed or is one jump from the frontier. Consider the top-left corner of the map as the obvious example: What possible reason do you have for NOT occupying Causeip and Ashteio? They are both safely behind your chokepoint defences, as is the Superterminal world on Qachar. Likewise, along the left edge of the map, you've got six hostile systems on or above the Ginwig-Ginyae-Chrsi line that are mixed in with yours and, if you took control of them all, could be defended directly by only two chokepoints, Ginya-Chrsi, while allowing you to reduce defences in three current chokepoints behind the line. If you take down the Superterminal, you can certainly afford it, and it will give you lots of knowledge. (And if you then take Weppan-Hartus-Neiner, you own the entire western side and have an enviable defensive setup with lots of space for the economic stations you've already paid to unlock, giving you only a few contacts with the enemy and the energy to build strong distributed defences in every system leading to your homeworld)
Fourth, regards to knowledge, one obvious thing to note is that you've skimped on turrets in favour of spending on your economy on the per-galaxy capped forts and shields. 28000 knowledge on your economy, but only 17,750 on turrets? Okay, I understand the Mil3 stations. I even understand the Econ3 stations, despite you controlling so little territory that isn't on the frontline and hence has military stations that you don't get much benefit from the economic stations (my solution would be to conquer more territory as noted above, taking down the Superterminal along the way), but regardless of that you don't control anywhere near enough territory for it to be worth additionally spending 9k on Harvester upgrades, and yet you've done so.
For that 9k price you could, for instance, have unlocked Needler 4, Laser 4, Missile 3+4 for 5500, and throw in Sniper III and IV for 3500. I daresay they'd help you more than your harvester upgrades, given that you are to a large degree living on salvage income anyhow.
Now,
I wouldn't unlock those if I only had 9k to use - I'd go primarily for Sniper, Spider, and Missile turrets, and I'll tell you why - I love defensive depth. Key to using defensive depth is knowing that it is never about
stopping the enemy before the final destination (in this case, your homeworld, later in the game a spire city), it is about inflicting attritional damage on the way, and THAT is mostly about maximizing time-on-target for defences. If you try to stop the juggernaut head on in each system before reaching the next, it will roll over you. Even worse, if it comes in multiple successive groups (which exowaves do) and the first group roll over your defences, the second wave may not even suffer attritional casualties when it is its turn to pass through the system!
Certain things follow: A system is intended for one of two roles, either stopping an enemy or inflicting attritional casualties on an enemy, and its defences should be set up accordingly, but in both these cases, Sniper and Spider turrets are invaluable. Exowaves consist of both large and small ships - ideally you'll mainly have to bother about the large ships while your defences kill most of the smaller, and few things are as good for weeding out the smaller ships as having sniper and spider turrets set up safely away from the enemy keeping their perfect 100% time-on-target damage for both of them, and slowing small ships in their tracks for the spiders.
Putting the station for the defences directly on top of the exit wormhole leading to your homeworld as you've done almost everywhere ensures that the station will be wiped out before the enemy progresses any further; Putting all your turret defences in a standard Kahuna-style defence ensures that they get wiped out too. This is inefficient. The purpose of a good defence in depth is to take advantage of that depth to bleed the opponent while limiting your own casualties.
Rather, for attrition purposes, the station should either be located as far as possible from the route the enemy is traveling (and be a logistics station) with all sniper and spider turrets placed far from the anywhere anybody passing through would come (e.g. behind the station), with other turrets placed to cover the route passersthrough traverse at maximum distance, OR be placed close by the exit wormhole but to the side of the line of advance, such that defences set up protect the station also overlap the line of advance to the wormhole but enemy ships are capable of traversing the line without having to fully destroy the station+forcefields. (The second should be your preferred for systems that double as current tanking locations with hostile wormholes).
In both cases, you may lose some or all of your defences to enemies that pass through, but a) it will take longer time for the enemy to do this, time in which it is under fire, and b) it is not guaranteed to wipe out your defences, which posting them directly on the wormhole does.)
MOREOVER, since the Fallen Spire exos arrive in several individual groups, NOT having them bunch up is a very good idea unless you expect to be strong enough to stop them cold in a system. Once your FS fleet is larger, you may very well end up with a Level 2 spire city that boosted with other defences makes an excellent chokepoint on which to KILL THEM ALL, but you don't have any such locations now.
Fifth, USE THE ZENITH TRADERS! You've played 14h and your number of toys is depressingly low. Your homeworld does not have the full set of obligatory Trader Toys - I refer to the Radar Jammer II, the Armor Inhibitor, the Armor Armor Booster, the Black Hole Machine... They should all be placed under a forcefield behind the home station, so they can live in peace and not be destroyed during construction should anything hit the home's forcefields. Nor have you built the Orbital Mass Drivers or the Superfortress. Sure, these toys are expensive, but they are powerful and if you cannot think of anywhere else to put them, the homeworld is also a nice location for those toys (and, in fact, with you having to defend on so many fronts, a
very good idea for defence in depth as exos coming from any direction will end up there after hitting only two systems). You've got the traders enabled, why on earth not make good use of them? Between the exos and the regular waves, you'll have more than enough metal income to build their toys. (If you feel you have too low a metal income,
just increase your AIP by conquest and the increased wavesize will do the rest).
Anyhow, I hope there's something in what I've written that will be of use. Much of my frustration is because I simply don't see the logic of the way you've expanded and spent knowledge, apart from a vague feeling that you've been focusing on limiting your AIP expenditure over setting up a good defensive and economic network.
While it probably makes sense to you, and it may even make good sense, as people have different playing styles and I'll not insist that my own is the best after having played the game for so short a time (2 months), when I look at it I only see all the great missed opportunities for setting up secure lightly defended inner areas that pay for significant defence in depth and major defences on the borders. You've focused on unlocking defences with per-galaxy cap and spread them over many systems rather than focusing on those with per-planet caps, with the results that - as I see it - you are weak everywhere. Forts are fun and I highly recommend them, but they don't provide anywhere near the bang for the knowledge-buck that turrets do when you are defending more than one planet.
The problem is, as I am seeing these things through my strategic vision and not yours, I am not sure that my advice is all that useful to you in the given situation unless you are willing to depart from your own and adopt mine, and since you cannot take back actions already taken, that will require you to start killing off a bunch of enemy systems, even those nasty eyes in the regions I advocate clearing first, to afford the defences I'd already have in your place.
(Fortunately, eyes are fairly easy to kill. Parasite Eyes don't hurt starships, so if you have good starships, and you do as you have Spire Corvettes, taking those down should be trivial (or would be if you unlocked higher tier Spire Corvettes and higher tier sniper turrets). Plasma Eyes can be nasty against Starships, but they provide overkill vs fleetships and can only target 10 per 2s, so just send in the whole fleet and accept the casualties. Threating Eyes threaten, and who cares, that's just more chaff. Nuclear Eyes should be hacker-sabotaged, and sabotage is, of course, a completely acceptable solution to
any Eye threat.)
EDIT: It is tempting to think of Mil3, boosting strength by 100%, and Log3, reducing enemy speed by 50% as roughly equivalent. Nothing could be further from the truth. It depends entirely on the firing range of hostile ships, the location within the gravity well of the station, and on whether the time that is won from the use of logistics stations is put to good use elsewhere. And since time is the single most valuable currency in most strategy games, that should be possible. One classic example: First defending one planet with ALL HBC turrets backing up its regular defences, then as an exo passes that planet and continues into a logistics corridor, tear down any surviving HBCs and build them in the next planet on the line of advance, until they are in the final system. In case of limited resources, just do this with HBC IIIs and IV and leave the others in the system where you expect greatest time-on-target.)