Restricted version of vertex cover
Let us call your problem as $\mathsf{RESVC}$. $\mathsf{RESVC}$ is NP-Complete. 


Given a subset $S \in L$ its easy to verify in poly. time whether it forms a
valid cover of size k. Hence, $\mathsf{RESVC} \in \mathsf{NP}$
$\mathsf{RESVC}$ is also NP-hard because the decision version of the Minimum Dominating set problem reduces to it. And, we know that the decision version of Minimum Dominating set is NP-complete. The reduction goes as follows:

$\qquad \displaystyle \mathsf{D\text{-}MIN\text{-}DOM\text{-}SET} = \{ \langle G,k \rangle : G \text{ has a dominating set } S \text{ s.t. } |S| \leq k \}$

Reduce the instance $\langle G,k \rangle$ of above language to $\langle G',k \rangle$ as follows:

Lets say $G = (V,E)$ and $V = \{v_1,v_2,\dots,v_n\}$ and define $G' = (L \cup R, E')$ with


$L  = \{v_{11},v_{21},...,v_{n1}\}$  
$R  = \{v_{12},v_{22},...,v_{n2}\}$  
$E' = \{(v_{i1},v_{j2}) : (v_i,v_j) \in E \vee (v_j,v_i) \in E\} \cup \{(v_{i1},v_{i2})\}$


Now, it is easy to see that $\langle G',k \rangle \in \mathsf{RESVC}$ if and only if $\langle G,k \rangle \in \mathsf{D\text{-}MIN\text{-}DOM\text{-}SET}$.