In this chapter, we present necessary and sufficient conditions of finiteness and upper bounds for power and exponential moments for some modifications of Markov generalized hitting times, namely extended Markov generalized hitting times. These upper bounds are obtained using the method of test functions. The extended Markov generalized hitting times appear in applications to semi-Markov processes with distributional atoms, semi-Markov processes admitting artificial regeneration, and multi-alternating regenerative processes modulated by such semi-Markov processes, where they play roles of regeneration times. The moments of extended hitting times are key in getting explicit upper bounds for convergence rates in ergodic theorems for such processes. We derive integral equations for moments of extended Markov generalized hitting times and obtain serial representations for these moments as minimal solutions of these equations, present necessary and sufficient conditions of finiteness, and give recursive upper bounds for power moments and upper bounds for exponential moments of extended generalized hitting times.