Green’s Dictionary of Slang

hip n.1

[SE, but note also hip, the n.]

(US Und.) a burden, a problem.

[US]Jackson & Hellyer Vocab. Criminal Sl. 44: hip [...] A burden; an attachment; a responsibility; an incubus.

In phrases

on one’s hip

1. pressurising, harassing.

[US]C. Mathewson Pitching in a Pinch 112: McGraw is always on Devore’s hip because he often feels that this brilliant young player does not get as much out of his natural ability as he might.

2. of a person, acting as a burden, a dependant.

[US]Jackson & Hellyer Vocab. Criminal Sl. 44: I can’t see you tonight; I’ve got a Jane on my hip.
[UK](con. WW1) Hall & Niles One Man’s War 192: If those Britishers couldn’t be stopped before they dropped their bombs, we would have Switzerland on our hip.

3. having someone following one.

[US]Jackson & Hellyer Vocab. Criminal Sl. 44: Don’t round, we’ve got somebody on our hip.