The transfer simply minutes earlier than the pair backed amendments geared toward toughening up the flagship laws.
In all, round 40 Tory MPs are anticipated to defy the prime minister in a serious blow to his authority.
Anderson, the controversial MP for Ashfield, was appointed to the function in February final yr.
Fellow Pink Wall MP Clarke-Smith solely took up the put up two months in the past.
Saying his choice to help amendments within the title of former immigration minister Robert Jenrick and Invoice Money, he mentioned: “Once I was elected in 2019 I promised my constituents we might take again management.
“I need this laws to be as robust as attainable and due to this fact I shall be supporting the Jenrick/Money amendments. These are arguments I’ve persistently made and can proceed to make.”