Hi there mattclaydon-hotm,
It's vital that all your transactions are recorded correctly in your books for tax purposes. QuickBooks Self-Employed allows you to import transactions from the bank in order for you to easily categorize them in your books. I can provide some information about expenses in QuickBooks Self-Employed.
There isn't any way to specifically enter expenses that are meant to be billable to a customer. What you could do once the expense comes through would be to create it as a regular expense and then create an invoice in the system to send to the client.
If you have any other questions, feel free to reach out here.