Not a good idea if your purpose is to 'win' back his heart with gifts or to compete with the OW.

If it's just a non-romantic gesture of acknowledging his birthday then I suppose a card will do.

One last thing, why do you think you need to compete with anyone?

M35 XW34
D5 D4
M 6years T 10years
Bomb 5/2013
Joint Petition signed 6/2013
Moved out end of 8/2013
Court Hearing of Joint Petition 9/2013
D finalized in 3 months - no news yet