We investigate the complexity of CONSTRUCTIVE CONTROL BY ADDING/DELETING VOTES (CCAV/CCDV) for r-approval, Condorcet, Maximin, and Copeland^α in k-axes and k-candidates partition single-peaked elections. We prove that CCAV and CCDV for most of the voting correspondences mentioned above are NP-hard even if 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. Additionally, we give a polynomial-time algorithm for recognizing 2-axes elections, resolving an open problem.