Continuous gravitational waves from neutron stars could provide an invaluable resource to learn about their interior physics. A common search method involves matched filtering a modeled template against the noisy gravitational-wave data to find signals. This method suffers a mismatch (i.e., relative loss of the signal-to-noise ratio) if the signal deviates from the template. One possible instance in which this may occur is if the neutron star undergoes a glitch, a sudden rapid increase in the rotation frequency seen in the timing of many radio pulsars. In this work, we use a statistical characterization of the glitch rate and size in radio pulsars to estimate how often neutron star glitches would occur within the parameter space of continuous gravitational-wave searches and how much mismatch putative signals would suffer in the search due to these glitches. We find that for many previous and potential future searches continuous-wave signals have an elevated probability of undergoing one or more glitches and that these glitches will often lead to a substantial fraction of the signal-to-noise ratio being lost. This could lead to a failure to identify candidate gravitational-wave signals in the initial stages of a search and also to the false dismissal of candidates in subsequent follow-up stages.