Fuel purchased by business on behalf of client in QBO?

If I interpreted your question correctly, the business paid for something (fuel in this case) on behalf of a client, who then repaid the business for the full exact amount sometime later? Perhaps the business even passed on the bills or receipts to the client, which could be made out in the client's name?

If that's the case, it's an IOU, advance or loan posting to a debtor account in the customer's name, awaiting future deposits to clear off when the client repays the IOU. There should be no raising of invoices, no claiming of VAT and nothing that would affect the P&L.

No. Without assuming a reimbursement from the customer---what account would you record this expense?

I have been using a separate expense account and labeled it "job supplies" since this expense is never known ahead of time and is spent for a specific job. These expenses tend to occur when customers drop off their vehicle for repairs that take some time to complete and upon drop off, the vehicle has very little fuel in it. Therefore, when the business has to test drive said vehicle it is left with absolutely no gas, so then the business replaces the small amount of gas (may replace with a bit more than when dropped off, anywhere from about $9-$20).

Does the account I'm using suffice? Or should it be part of COGS?