The series of molecular signals initiated by binding of a Wnt protein to a receptor on the surface of the target cell where activated receptors signal via downstream effectors including C-Jun N-terminal kinase (JNK) to modulate cytoskeletal elements and control cell polarity that contributes to the shaping of the heart. [ GOC:mtg_heart http://www.ncbi.nlm.nih.gov/pubmed/16860783 GOC:dph ]