In general, your card's required minimum payment will simply be a fixed percentage of your owed balance. Usually at a level just above the card's APY divided by 12. That'd be enough to cover that month's interest charge plus a (small) amount left over to pay down the balance. There's usually a "floor" of $25-35 that the required payment always stays above, but with a $10k balance you're probably a ways from that mattering.

I'd expect the required minimum to be set in the card agreement and be non-negotiable. But it might be worth calling the card and asking, you might get lucky. Though be aware that even if they allow that, if your payment is lower than your monthly interest charge then your balance will increase month to month, even with no new charges.

Honestly, the only real way out is what you appear to already be doing: get income larger than spending to have more money available to pay down the card balance faster.

But a few potential short-term bandaids:

• Do you have the ability to make any more cuts to spending? It sucks, but if you focus on it being only until the card is out of your life, that can make it easier. "Eat out less", "make coffee at home" are cliches, but can help, if only a little. But every little bit helps.
• See if you can be approved for 0% transfer card. Moving this $10k over to that would remove the interest you're paying on this balance each month. Those transfers do tend to have a fee, usually a percentage of the balance. So you may have to pay $300-400 to get that lower rate, and it'll only last 12-18 months.
• Borrow money from elsewhere (personal loan, friend, family) to zero out this card and pay that other loan back at a lower rate.