be tied to (something's) apron strings
To be controlled or dominated by something. If you accept money from that unsavory organization, you'll be tied to its apron strings for a long time. The CEO of that company is a big political donor, so of course he's tied to his party's apron strings. Once your book comes out, its tied to that publisher's apron strings. If you're having serious misgivings about them already, I'd look for a new publisher.
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