Can I widen the hole with a drill to straighten the handle?
I would plug both holes without enlarging them, glue a piece of dowel that fits and let dry.
Then check on the face that the handle will cover the dowel when the handle is mounted correctly, otherwise you will need to make it the same color.
Then drill new holes in the correct positions. The back side may be easily covered by using a washer or a screw with a wider head - usually the screws for those handles have a washer built into the head anyway.
The adage: measure twice, cut once also applies to drilling.