根に持つ literally means ‘have (something) at the root’. Figuratively it means you hold on to a bad memory about someone for a long time. A similar English expression is probably “to hold a grudge (against someone)”, but the tone of “根に持つ” can be lighter, more trivial and less serious than that.
（〜を）根に持つ “hold a grudge (against/about ~)”
It’s often used as “根に持っている” meaning “be/have been remembering/holding a grudge”
I still remember (and am annoyed) that he ignored my message three years ago.
恨みがある: hold a grudge (sounds more serious than 根に持つ)
引(ひ)きずらない: don’t dwell on ~ (in the past)
すぐに忘(わす)れる: forget soon/easily
related post: Funny Japanese Idioms about Plants