You get a French social security number (and card) when you start to work. But normally you and the children should be included on your partner's social security card. Does he have a French social security number/card?

If he does, he should contact the social security office and have you and the children added to his card. Then, you take his card with you when you go to the doctor. You still pay in full, but then he is reimbursed into his bank account by the sécu (for 2/3 of the amount) and by whatever mutuelle he is covered by (assuming you and the children are covered by the mutuelle) for the remander, less one or two euros.

Otherwise, you have to claim against your UK coverage and/or the european health insurance card you mentioned.

There are special rules in most countries for those folks who live in one country and work in another. As I understand it, your partner should be paying into the French health system if he has declared his home to be in France. But he should talk to his employer to see how that works between the UK and France. (I'm only familiar with the France-Germany rules but was told it works in a similar manner throughout most of the EU.)
Cheers,
Bev