Can my free hand touch the table when hitting a ball in play in table tennis?
No, then you lose the point.
According to ITTF's Rules (The International Table Tennis Federation's Handbook 2019 v2):
2.10 A POINT
2.10.1 Unless the rally is a let, a player shall score a point
18.104.22.168 if an opponent's free hand touches the playing surface;