I love the dramas where the ML has a kid, and the kid ends up loving the FL as well. 

I've seen unforgettable love. And I'm currently watching Love's Ambition, with this in the plot as well. Any other dramas you can think of? 

One Spring Night (one of my all-time favorites)

Love Scout (also great, very refreshing with pretty reasonable, mature characters)

Lie after Lie though the kid is actually her kid, who got taken away when she went to prison.

Story of Minglan, ML has two kids. 

Eternal Love - the kid is actually the FL's too but she doesn't remember because of a big story thing that happens

+1 to both One Spring Night and Please Be My Family. I really liked them both. One Spring Night though is especially excellent.  

This might not qualify, but both leads have kids. It's older but so fun while being realistic, funny, but heartfelt and both leads are on the mature side. I don't want to say too much but it's a drama I think highly of. 

Five Enough

I also loved One Spring Night & Please Be My Family