I recognize the challenges you're facing with categorizing expenses and income correctly in QuickBooks Self-Employed (QBSE), userlh. It's crucial to accurately reflect these numbers in your records.
If you've paid the expenses on behalf of your customer, you can bill them for those expenses using the Billable Expense feature. However, it's only available in QuickBooks Online (QBO).
Since you're using QBSE, I recommend consulting your accountant to help you properly record these expenses. They can help you choose the correct account and bill your customers accordingly.
I've added this link about categorising transactions in QuickBooks: Categorise transactions. It also organises your income and expenses so you know what areas of your self-employed business have the biggest impact.
Furthermore, visit this article to learn the basics of QBSE: QuickBooks Self-Employed Overview.
Please don't hesitate to get back on this thread if you have other concerns about categorising expense transactions. I'm always glad to help in any way I can.