Hi John-Atherton,
What I'm getting here is you pay the business expense (mileage) using your private account, and you want to be reimbursed. Then, that expense is also billable to your customer. I'm happy to walk you through the process.
First, you can create a journal entry to record the business expense against your equity account. You use an equity account since it's from your private account. Please have this checked this with an accountant.
Second, you reimburse your private account through a cheque or an expense transaction. Then, make it billable to your customer. This is where the billable comes in since we're unable to do it on the journal entry.
Here's an article with the detailed steps: How to pay for business expenses with personal funds.
If you have other questions, feel free to post back.
