Approximation algorithms for Directed Minimum Cut with Cardinality Constraints
We can get a $(2,2)$ bi-criteria approximation as follows (or more generally $(1+\varepsilon, 1 + 1/\varepsilon)$ bi-criteria approximation).

We may assume that we know the cost of the optimal solution. Denote it by $OPT$. Let 
$$w'(u,v) = \frac{w(u,v)}{OPT} + \frac{1}{k}.$$
Consider the optimal solution $(V_1, V_2)$. Then
$$\sum_{(u,v) \in E(V_1, V_2)} w'(u,v) = \sum_{(u,v) \in E(V_1, V_2)} \left(\frac{w(u,v)}{OPT} + \frac{1}{k}\right) = 1 + \frac{|E(V_1,V_2)|}{k} \leq 2.$$

Our algorithm finds the minimum directed $s$-$t$ cut $(V_1', V_2')$ in $G$ with edge weights $w'$. The cost of this cut is at most $2$. Therefore, the cut $(V_1', V_2')$ cuts at most
$$E(V_1', V_2') = \sum_{(u,v)\in E(V_1',V_2') } 1 \leq k \sum_{(u,v)\in E(V_1',V_2')} w'(u,v) \leq 2k$$
edges. The cost of the cut is at most
$$\sum_{(u,v)\in E(V_1',V_2')} w(u,v) \leq OPT \sum_{(u,v)\in E(V_1',V_2')} w'(u,v) \leq 2OPT.$$