Function-Correcting Codes (FCCs) are a novel class of codes designed to protect function evaluations of messages against errors while minimizing redundancy. A theoretical framework for systematic FCCs to channels matched to the Lee metric has been studied recently, which introduced function-correcting Lee codes (FCLCs) and also derived upper and lower bounds on their optimal redundancy. In this paper, we first propose a Plotkin-like bound for irregular Lee-distance codes. We then construct explicit FCLCs for specific classes of functions, including the Lee weight, Lee weight distribution, modular sum and locally bounded function. For these functions, lower bounds on redundancy are obtained, and our constructions are shown to be optimal in certain cases. Finally, a comparative analysis with classical Lee error-correcting codes and codes correcting errors in function values demonstrates that FCLCs can significantly reduce redundancy while preserving function correctness.