"To offer a bonus" means to provide or give an additional reward, typically in addition to a regular payment, as an incentive or reward for something. Full definition
Many cash back cards offer bonuses for certain types of purchases and a lower level of return on everything else. (mybanktracker.com)
Many cards offer bonus points for spending in certain categories, i.e. spending at grocery stores, or for meeting a certain spending requirement. (travelandleisure.com)
Some travel credit cards have no bonus at all, but the best travel rewards cards offer bonuses of up to 100,000 miles or points, sometimes worth thousands of dollars in free travel. (investmentzen.com)