Percolation theory provides a quantitative framework to estimate and enhance robustness of complex networked systems. A typical nonstructural method to improve network robustness is to introduce reinforced nodes, which function even in failure propagation. In the current percolation models for network robustness, giant connected component (GCC) is adopted as the main order parameter of macroscopic structural connectedness. Yet there still lacks a systematic evaluation how mesoscopic network structure evolves under measures for network robustness. Here, on networks with reinforced nodes, we study K-core structure, which consists of nodes with degrees K and plays a vital role in structural stability and dynamic spreading. We develop an analytical framework for K-cores on uncorrelated random graphs with randomly distributed reinforced nodes. We verify our analytical prediction with simulation on generated random graphs, and show that, with a set of reinforced nodes beyond a critical size, an abrupt emergence of K-cores is smoothed out to a continuous one. Our framework offers a controllable way on tuning sizes of K-cores to explore their various roles in network structure and dynamics, besides designing robust systems.