Uncomfortable riding position

I have a road bike with dropped handle bars. I feel fairly comfortable on
it and it feels most comfortable to position my hands on the corner of
the handle bars. I do sometimes place them at the side but not for long
periods of time, it feels a little bit like I'm stretching. I don't put
them on the top because I have the brackets for my lights there.

I've noticed that when I do push ups it's very painful on the palms of my
hands and I think the way I ride my bike probably had caused this.

Could the way I position my hands suggest that the size of the bike is
wrong or something needs adjusting?