  We investigate the complexity of {\sc{Constructive Control by Adding/Deleting
Votes}} (CCAV/CCDV) for $r$-approval, Condorcet, Maximin and
Copeland$^{\alpha}$ in $k$-axes and $k$-candidates partition single-peaked
elections. In general, we prove that CCAV and CCDV for most of the voting
correspondences mentioned above are NP-hard even when~$k$ is a very small
constant. Exceptions are CCAV and CCDV for Condorcet and CCAV for $r$-approval
in $k$-axes single-peaked elections, which we show to be fixed-parameter
tractable with respect to~$k$. In addition, we give a polynomial-time algorithm
for recognizing $2$-axes elections, resolving an open problem. Our work leads
to a number of dichotomy results. To establish an NP-hardness result, we also
study a property of $3$-regular bipartite graphs which may be of independent
interest. In particular, we prove that for every $3$-regular bipartite graph,
there are two linear orders of its vertices such that the two endpoints of
every edge are consecutive in at least one of the two orders.
