If your client gave you a check, you'll want to record that check payment on the relevant invoice (click here to see how).
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Once you've either received an online payment via Ivy or recorded a payment manually, you will be able to send a receipt for that payment. You can do so by following these instructions:
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Click "See Details" in the Payments section to the right of the invoice to reveal the Applied Payment(s). Under the payment record you want to send a receipt for, you should click on the icon that looks like this: 

Clicking on this icon will give you a pop-up window where you can enter the email address(es) that you would like to send the receipt to. Once you've entered the email address(es), click "Send" and you're all set!
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You can also re-send the invoice to your client at any time and they will be able to see the same "see details" option in the Payments section like you do, so they will always have access to see any applied payments there.