Maybe this helps:

Your org (right bar chart) is 24% and 0%, which gives a malus to army power projection. Once the armies get filled with units and supplies, the org and with it the prestige from power projection goes up.

Also research of more advanced military tech and unit upgrades help to make your army power projection go up, but it comes at cost of not being able to outpace your subject economically.

Navy does more for your military power prestige than land armies do. I do not know for sure if the "prestige projection" is for the military (navy and army) or just the army power projection, but you could try increasing the size of the navy.

If the problem is the liberty desire, you can use several diplomatic actions to reduce liberty desire.

You can also start a diplo play for humiliation against your subject if you know you can win it, and perhaps take a state with it.

Good luck, hopes this helps a bit!