Expenses refund from customer

I’m contracting through an agency for large company X. I spent a few hundred £ on a trip for X over about 5 different transactions. I then had to submit an expense claim to the agency, for them to pass on the costs to X. The agency then paid me as a single line item on the same self-bill invoice as my weekly hours claim.

To complicate things they added VAT to the expense claim, so the following week had to deduct the VAT from the next self-bill invoice.

I’ve previously managed to get my head around how to account for making an expense claim with multiple items if I’ve paid for them from my personal bank account and I’m claiming from my own company. In this case my company is claiming back from a customer and I’m not quite sure where to start with both the vat thing and the effective pass through of costs? Any guidance would be gratefully received (I’ve done a search but not found the answer I’m after!)