When creating a payment in Core, is it possible to search by invoice number in the Add Payment screen?
Yes, you can do that by choosing the By invoice number option for Find Invoices in the Add Payment dialog. It allows you to search and select an invoice for payment. Check the Core Help for details.
An alternative to this is to make payments on invoices via the Invoices screen directly. Search for the relevant invoice there in the list view, then click on its row-action menu and select Create Payment. It takes you to the Add Payment screen with the relevant invoice selected. Check the Core Help for details.