Your situation at the moment is incredibly generous I'm afraid, and childcare does not fall within expenses.
Expenses are the direct out of pocket costs of doing your job - travel, food whilst away, etc. It does not extend (and I'm not trying to comment on the feminist question of whether it should) to knock on costs you incur because of your personal or home life.
Your DH's employer won't be able to pay childcare costs as expenses, I shouldn't think. If they do, I think it's a taxable benefit, so paying it as a simple refund would infringe tax rules [disclaimer- am not a tax lawyer!].That's where displacement or other allowances come in, because I assume that, if those are designed to cover more than the direct costs of travel, food, etc, they are put through PAYE on the balance.
Look at it from the viewpoint of a single parent. Would you expect the employer to pay all childcare because the employee couldn't work without it? What you are asking for is essentially a similar request.
Sorry, I know that that won't be what you want to hear.