I'm Just adding to Annatar
If you actually see superweapons, that means you better either have a booming economy or a massive fleet. If SW's are out, that means big trouble that's going to be difficult to counter purely because of the logistics of getting to the superweapon to counter it.
Novaliths
If you're playing entrenchment, starbases have an ability (lower right hand corner on the box selection screen I believe where you upgrade the starbase) where it'll be called something along the lines of "undying allegiance". No matter what kind of damage is done to the gravity well, as long as the starbase stands, you will keep your planet, even at 0 health.
That's a very very expensive option. Won't always be prudent.
Vasari gets an upgrade in the first tier of the econ tree where planet health upgrades are 50% cheaper. That should help if you're Vasari.
Advent get up to 60% discounts on colonizing (with lvl 3 colonize on progenitor mothership). This isn't always the best option, because early game you don't really want to spend your resources on planet health (unless you're already in an early game fight somehow).
TEC scouts get a bomb ability they can use on buildings. It's in the third tier of the military tree. If somehow you can manage to get them back to the novalith, about 20 or so TEC scouts can destroy a novalith, or any other superweapon for that matter.
Kotsura
This one's probably the hardest to counter, but unless someone surprises you with 6-8 of these, I don't think you have much to worry about. The catch22 with the kotsura is it doesn't really do that much damage to a fleet. What is it? 250? 500? It has to go through Mitigation shielding, which really takes the oomph out of it. Especially if the kotsura user doesn't stagger fire them (fire one, mitigation shields go up, wait, mitigation shields go down, repeat). The Kotsura is much better at taking out buildings (except for starbases).
Most effective option is to split the fleet. The problem is that the Kotsura (which is the best superweapon imo) takes down the only thing that can destroy it. A fleet is the only thing that will ever destroy a Kotsura.
Deliverance Engine
This superweapon is much more of a tactical option for making the Advent fleet better. This goes in the complete opposite direction of the other two. Novaliths and Kotsuras destroy. DE's make the Advent fleet better. When you fire a DE, advent fleets get tougher (additional 6% mitigation) and get stronger (do 25% more damage). Not to mention the culture lingers.
Best option against the culture is to get media hubs/broadcast centers/etc. and place them pretty consistently throughout your empire. When you're fighting DE, this is actually the only time I've ever researched the tech in the culture tree that gives you 20%/40% resistance to enemy culture.
As long as you can keep consistently making your own culture you should be able to outlast the effects of the enemy culture (culture killing takes forever). You will need to destroy the DE eventually, but then again, that's what it's like with all super weapons.