How to fix a nail that is breaking around the side?

If you are referring to the head of the nail breaking around the center of the nail,...
it happens due to mfg…
best thing is to take a hammer and a nail punch and just tap in in below the surface of the wood, and fill it in with wood filler, like spackle.

there is glue you can buy that is specifically for fixing nails (real or fake ones!)
:)
I haven’t had much luck with it because both of my hands shake and it’s glue so there isn’t much leeway for mistake :)

Crazy glue works well.