The game only really explains the first one, and doesn't explain what counts as having done the thing you should do. That's step one, make sure it's clear in the in-game description of the activities both where you have to do the activity and where you have to hand in the progress, using the terms the galaxy map uses.
Perfect explanation of what's wrong.
In theory the galaxy map can assist with the second but because nothing in the game relates the terms to describe the different activities (Acquisition, Undermining, Reinforcement) to the system states the map displays, nor does it filter for validity for Acquisition.
It does, but split into two different places and there's an irritating UI behaviour.
You can find out what Power actors (including you) are already doing, using the left-hand menu. Choose the
second option in the submenu. It will reset the chosen Power every time you do it which is infuriating, but nevertheless, it works. You can see in my example a bunch of hollow stars on the lower right; those are systems Winters could acquire (and you can see there are other filters to choose.)
Click on the exciting new acquisition target! Then over on the right hand side you've got the Powerplay screen we've all come to love/hate. You can see right at the bottom I've got the mouse hovering over "Toggle Strategic View", that little globe...
When you click that you get the sphere with radial lines, and basically, if there is a radial line, the systems can influence one another.
Then - and this is especially stupid - you have to mess about remembering which kind of Power cargo does which for what phase and there are no clues to that on the Power contact screen.
Author's note: I am not acquiring Undini at this time, I'm well over the other side.
Systems valid for Acquisition (Unacquirable systems due to distance will currently highlight).
The left-hand screen does this. The right-hand screen gives you the per-system view to confirm. So you kind of can see both perspectives but it's clunky.
Systems with current Acquisition progress.
The "Minimum Activity Level" filter at the bottom of the left-hand screen helps in theory but it's badly calibrated so not that much use in practice.
Opponent systems with current Undermining progress.
You can choose any power on the left-hand screen and filter it on Undermining. You can also choose your Power and select Resist Undermining if you'd rather defend than attack.
Owned systems with current Undermining progress.
Choose your Power, choose Undermining.
That would let you go to the galmap and filter it to direct where you actually put your progress in order to advance your power.
The left-hand menu sort of does that, as above, and does have a nice intuitive left to right flow in thinking when you're using it to strategerize, so the principles of the UX are there. It's just as ever, the actual UI is needlessly clunky and full of papercuts.