The read tape in the definition of L and NL (logarithmic space) (DLOGSPACE, NLOGSPACE) (sublinear space)
For a complexity class $\mathcal{C}$, we can define read-$k$-times $\mathcal{C}$ to be the set of problems $S \in \mathcal{C}$ for which there exists a $k$ such that an accepting computation for $S$ on input $w$ reads each bit at most $k$. 

A consequence of a result from On lower bounds for read-$k$-times branching programs (1991) by Borodin, Razborov and Smolensky gives a separation between deterministic log space with unbounded reads and non-deterministic log space with bounded reads:

Proposition: There is a problem $S$ for which $S \in L$ but $S \not\in \text{read-}k\text{-times } NL$. 

This means, to answer your question, we lose some languages since $L \not\subseteq \text{read-}k\text{-times }NL$ but $L \subseteq NL$. 

My source of this proposition is from DynFO: A Parallel, Dynamic Complexity Class due to Patnaik &amp; Immerman, who cite the BRS paper.