I am assuming you are an aide. Well, it depends on your mistakes. If drug related, than you shouldn't be an aide. LPNs and RNs have lost their licences if found stealing meds from a patient or the facility. You need to understand that these agencies have a responsibility to their clients. Their employees go into private homes and they have to make sure the employees they send in have no priors. I wouldn't hire an agency that didn't do background checks. Most employers do them now to protect themselves and drug tests.
You may have to find another type of work.
The background check is one of the benefits of using an agency. People don't want past criminals working with vulnerable elders in their homes.
And even people who hire freelancers will frequently run background checks.
Perhaps your mother could pay you for caregiving? That wouldn't require a background check.