1.
If you ~ on something, you find out information about it.
It is certainly worth checking up on your benefit entitlements...
The Government employs tax inspectors to ~ and make sure people pay all their tax.
PHRASAL VERB: V P on n, V P
see also check-up
2.
If you ~ on someone, you obtain information about them, usually secretly.
I’m sure he knew I was checking up on him.
PHRASAL VERB: V P on n