Many optimization problems solved repeatedly in practice, such as in power systems or logistics, share a fixed structure but with different parameter values. Re-optimizing each instance from scratch can be computationally prohibitive, while adopting a single solution for all realizations is overly conservative. The K-adaptability paradigm offers a flexible compromise by precomputing a small set of candidate solutions and selecting the best once uncertainty is revealed. When K is small, this approach enables real-time decision making and provides human decision-makers with a manageable set of implementable solutions. However, it is often impossible to identify a small number of representative solutions that remain feasible across all parameter realizations. We introduce K-adaptability with recourse, in which the goal is to precompute K candidate assignments for the discrete variables while allowing the continuous recourse decisions to be optimized online after uncertainty is observed. This structure preserves online computational tractability and implementability while substantially improving adaptability. We present formulations of this model and develop heuristic and exact methods to solve these formulations. An extensive computational analysis compares the performance of the proposed algorithms on single and multi-dimensional knapsack and capacitated facility-location problem classes.