Child custody and child protection are gaps in libertarian theory
Libertarian theory doesn't really seem to address, "Who gets the kids when the couple splits up?" and "When can others ethically intervene to 'save' a kid they think is in danger of being abused by the parent, by taking that kid away"?
Can the marriage contract specify who gets the kid in the event of a split, or would that be unenforceable? Libertarians often believe that contract provisions should be regarded as unenforceable, e.g. if it's a slavery contract. Maybe a contract saying who gets the kid would be considered a slavery contract; I dunno.
A lot of libertarians also believe that you should be able to just rip up a contract you don't like and walk away from the deal, with no penalty other than that your property might get seized. An indigent woman who lives off of men's support might not care about that penalty much, because she doesn't have any property for anyone to seize. She's judgment-proof. So she could just take the kid and leave and get some other sugar daddy, as long as he didn't mind having some other man's kid around. Or in a libertarian world, maybe she'd live off of charity (since the government wouldn't be around to hand out welfare).
I've never seen libertarians address these issues.