When doing your push ups, when you rst you may be burning some of your energy, so don't do it all too often.
Everyone has their own way of doing pushups, but as you will learn, different types of pushups will work differen muscle groups. (close hand, regular, diamond, wide-arm)
Whe you do your pushups on the APFT your hands cannot leave the ground or mat. but, if you start to feel like you cannot go on, try to wiggle your hands out wider without them leaving the ground. This will allow you to focus your push-up efforts on different muscles and you may be able knock out a few more. Also, when you start to get tired doing that, wiggle them back closer.
This may help you knock out a few more, but it seems like you are getting to be pretty good at the push-ups alreayd if you are up to 20 +.
When you are practicing your push-ups and you absolutely cannot do anymore, drop to your knees and knock out some more. As you do more and more, you will start to increase the amount you do.
If you have not done so already, have the recruiters take a look at your form. I can knock out lots of push-ups if I use a crappy fom, but they won't count on the APFT.