They've only been dating for a few months and already she's 'got her claws into' him! [=she already has control over him]
With budget cuts looming, the government is trying to 'claw back' some of the money it promised to state agencies.
('chiefly Brit') The press has been 'getting its claws into' him. [=has been attacking/criticizing him]
(..) Company, halt! My lord. My lord. My friends, the hour has come. Madness has overtaken this city and grasped in its claws my children. But now we must drive it back under the rocks whence it came. Madness has had its day! Forward, march! Rich or poor, noble or common, (..)2016 TvShow: Game of Thrones Title: Blood of My Blood Season: 6 Episode 6