There are a lot of credit card offers for people with bad and limited credit and some offers even come with rewards programs. For example, you can choose Discover it secured card. This card requires a refundable security deposit, but allows you to earn 1% cash back on all purchases, including on Google store purchases. This card has no annual fee. As a bonus, Discover matches cash back you’ve earned at the end of first usage year. So, you may be able to offset your deposit by cash back rewards.
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