This problem was been answered in negative by M.Laczkovich (http://www.ams.org/journals/proc/1998-126-06/S0002-9939-98-04241-5/S0002-9939-98-04241-5.pdf). He constructed a proper Borel subgroup $H$ of the real line which cannot be covered by countably many sets $H_i$ with nowhere dense sums $H_i+H_i$. On the other hand, Laczkovich proved that each non-open analytic subgroup $H$ of a Polish group $G$ can be covered by countably many closed sets $A_i$ such that any non-empty closed subspace of $A_i$ contains two relatively open non-empty sets $U,V$ with nowhere dense sum $U+V$ and difference $U-V$.

This problem was been answered in negative by M.Laczkovich (http://www.ams.org/journals/proc/1998-126-06/S0002-9939-98-04241-5/S0002-9939-98-04241-5.pdf). He constructed a proper Borel subgroup $H$ of the real line which cannot be covered by countably many sets $H_i$ with nowhere dense sums $H_i+H_i$. 

On the other hand, Laczkovich proved that each non-open analytic subgroup $H$ of a Polish locally compact group $G$ can be covered by countably many closed sets of Haar measure zero.

Trying to generalize this result to non-locally compact groups, Laczkovich proved that any non-open analytic subgroup of a Polish group $G$ belongs to the sigma-ideal generated by the family $\mathcal F$ consisting of closed sets $A$ such that any non-empty open subspace of $A$ contains two relatively open non-empty sets $U,V$ with nowhere dense sum $U+V$ and difference $U-V$. Truly speaking, this result of Laczkovich is not quite satisfactory as each closed subset of $G$ containing a dense set of isolated points belongs to the family $\mathcal F$. Consequently, the $\sigma$-ideal generated by the family $\mathcal F$ coincides with the ideal of meager subsets of $G$.

But we can ask another problem: can each non-open analytic subgroup of a Polish Abelian group be covered by countably many closed Haar null subsets?