Currently, applying a part payment to an invoice in QuickBooks Self-Employed (QBSE) isn't an option. This is because the invoicing feature in QBSE isn't part of the ledger and on your tax return.
You can create 2 or more separate invoices to record each payment as an alternative. Then, after you receive the invoice payment, you'll have to mark it as paid. You can refer to this article's Step 3 section for the detailed steps: Create invoices in QuickBooks Self-Employed.
Once done, you're able to manage your open and completed invoices on the Invoices menu. This is where you can edit, duplicate, print, export them as PDFs, and send reminders for overdue ones.
Additionally, you're able to customise your invoices in QBSE. You can have one set of customisations that apply to all invoices. For the step-by-step guide, you can refer to this article's Customise your invoices section: Invoice preferences.
Please know that you're always welcome to comment below if you have other concerns or follow-up inquiries about managing your invoices in QBSE. I'm just around to help. Take care always.