Some very dodgy or impressive timings here. My top tip with any phys plan is to always aim to become stronger.
Once upon a time I attended UKSF selection. Don’t worry- I failed!! But a key point I picked up on was to improve yomp/tab ability, you must get stronger! When you add strength, the less the load taxes your 1RM, allowing more energy reserves for subsequent tasks post March.
The same concept applies to press ups. The stronger you are, the less the load (in this case - your body) taxes you, and therefore you can move it up and down more times.
My advice therefore, is to add load to your press up or turn onto your back and get stronger by bench press and complimentary secondary exercises. There are minimal differences in kinematics and muscle activation between the two exercises. With the bench you can safely add more load and work in percentages which contribute to strength improvement. The core must contribute significantly as the load is increased. If your 1rm bench press is 100kg, then you can bench 50kg more times than someone with a 70kg 1rm. So if 2 people weigh 80kg, both with the same training age and technique (comes with time and also important) the stronger person will be able to perform more press ups as their ability to move 80kg will tax them less. Hopefully that makes sense. In other words aim to get strong.
When it comes to phys keep it simple and do the basics well.