By default, part payments on an invoice aren't shown. This means that unless the invoice is fully paid the invoice total due will be the full amount of the invoice. This can be easily changed in your invoice options.
Change this by going to Settings > PDF Themes > Your Theme > Misc Options, then here you will need to tick the box 'Show Payments' followed by 'Update'. Invoices will then show all payments. As well as emailing your customer a copy of the invoice with the part payment(s) displayed, you can also email them a remittance advice.