The maximum you can claim over 10000 miles is only 25p per mile; presumably you have made a typo.
You have no entitlement to travel and subsistence expenses for what amounts to residence to place of duty activities. Any such refund is taxable at your marginal rate.
I am astounded at such a tax treatment. It is little wonder that HMRC view umbrella companies with the deepest suspicion. The residence to place of duty element should pass through the PAYE system as income.
You are correct in your surmise. As an employee you may only have one place of employment and any travel therefrom on business may be refunded outside the tax system.
I simply have no notion what the umbrella company is doing and I am not surprised that you do not understand; nor do I!
You must take the whole matter up with them, it appears to be accountancy of the Mickey Mouse and Donald Duck variety which should not be tolerated for a moment. You are in danger of tax under deductions and losing genuine travel and subsistence refunds into the bargain. The umbrella company may think they know what they are doing, but neither you or I have any idea. I would be inclined to sharply get this sorted out and if you continue to receive woolly responses then sue for the sums withheld in the Small Claims Court.
That should concentrate minds in the umbrella company somewhat as there are no costs awarded in such hearings and it will probably cost them 4 figures just to put in a defence. Actions for underpaid wages are invariable successful and all but incapable of defending. I know, I have done one once.