CPAG (Child Poverty Action Group) have been challenging various aspects of the Two Child Limit Regulations. taking the matter all the way to the Supreme Court.
CPAG has contended that the Two Child Limit unlawfully discriminates against a number of different groups including, but not limited to: children, children with multiple siblings, large families and those with a religious or moral objection to the use of birth control. Further, they content that the principal policy justification for the limit is logically flawed. They believe that it is a policy which encroaches upon very personal and intimate decisions about family size and planning and treats some children as less deserving of a benefit intended to meet their basic needs purely because of their birth order.
This would appear to cover any family affected by the Two Child Limit although the two families involved in the appeal are lone parents.Unfortunately on 15 May 2019 the Supreme Court rejected a challenge by lone parents with young children to the reduced benefit cap, holding by a majority of 5-2 that its discriminatory effects are justified.
R (DA & Ors) v Secretary of State for Work and Pensions; R (DS & Ors) v Secretary of State for Work and Pensions  UKSC 21.
Lord Wilson stated: "The appellants have not entered any substantial challenge to the government’s belief that there are better long-term outcomes for children who live in households in which an adult works. The belief may not represent the surest foundation for the similarity of treatment in relation to the cap; but it is a reasonable foundation, in particular when accompanied by provision for DHPs which are intended on a bespoke basis to address, and which on the evidence are just about adequate in addressing, particular hardship which the similarity of treatment may cause." 
Previous case - kinship carers.
In SC and Ors v SSWP  EWCA Civ 615 - the Judge decided that the way the Two Child Limit worked for kinship carers was unfair (with the result that the government subsequently changed the rules); but that, as a whole, the limit was not unlawful.