In the US your credit card has a monthly billing cycle. After the end of each billing cycle, Payhawk would initiate an ACH pull from your linked business bank account. Top-Ups may be initiated via the Payhawk Portal.
In Europe you have your own dedicated account number and sort code in GBP, and international IBANs for GBP and EUR. You can deposit funds to your account like with any traditional bank account using SEPA, Faster Payments or SWIFT. Likewise, you can also receive funds from third parties.