Approximation algorithms for an instance of the Monotone circuit satisfiability
If this is a practical problem, rather than an approximation algorithm, I suggest using an off-the-shelf SAT solver.  There are standard ways to encode the constraint that at most $k$ variables are set to true, using an at-most-$k$-out-of-$n$ constraint.  For instance, Z3 has good support for this kind of constraint.
My impression is that approximation algorithms for SAT-like problems tend to perform very poorly.