Do I report my misogynistic supervisor to HR?

Hi, I am in my early 20's (21yo) and my supervisor (mid 40s) has been constantly harrassing me, spreading rumors, lying about my actions toward him or others, and purposely leaves me to do the work by myself. It's gotten so bad to the point that my hiring manager has put us onto seperate shifts so we don't cross paths, but the supervisor still tries to lie about me to my other managers to attempt to get me into trouble.

I believe I can claim hostile work environment due to discrimination against my gender but alot of what I have documented down, happened last year continuously, and I do have an occurence documented as recently as this month. Can I report him to HR and could I include the other times that my supervisor has harrassed me or attempted to lie about me in the HR report?

This is my first time having dealt with a hostile work environment but it has been constant, and I am constantly walking on egg shells but any interaction with my supervisor he tries to make himself out to be the victim.