Strengthening your core muscles will help strengthen your body overall. Although I'm not sure it will help you do more push-ups, it will help you overall. To get ready for the snowboarding season, I try and strengthen my core - this helps me stay out on the slops longer.
Are you attempting to do full body pushups or a modified version? Although concentrating on the core and legs can help, there are a number of other tactics to do as well. Just as we need to crawl before we walk, and walk before we run, how about starting with 'modified push ups' (on knees) and trying 5 of them. If your goal is to do 10, then add one per week, and in about a month, you should have accomplished your goal. Drawing the navel in, keeping your back in neutral, keeping your wrists directly under shoulders and your head/neck in neutral as well can all help. Let me know how you're doing.
if you concentrate on tightening your abs and quadriceps and the front of you shins (your tibia), then yes, you will be able to do more pushups. it is called "irradiation." Also tightening the rest of yor body will provide you a solid body to push up. That's easier than pushing a body that's sagging.
i'll give you an example: what's easier to lift? 50 pounds of a solid dumbbell or 50 pounds of a sandbag? Obviously the dumbbell cause it's solid, it has a handle, whereas the sand bag's weight shifts around when you move it.