When using a card processor, payments to your bank account are batched and/or with fees removed. (I'm using Stripe - other providers may differ)
When a customer pays a Kashflow invoice using the link to your card processor, that payment then gets attached to the invoice, and an entry is made to the bank account you set up in the settings for Stripe.
However, what the Knowledge Base article doesn't tell you, is if you use your current account you can no-longer reconcile it! The payment amount is not adjusted for fees, and it gets batched into a (daily/weekly/monthly) payment anyway, which is no good for your bookkeeping.
The solution is to add a new "Bank Account" in Kashflow, called (in my case) Stripe, and have the payments go there.
Such a simple solution that would help others, but Kashflow support didn't seem too interested, and directed me to here.