An efficient foraging strategy is vital for all living beings. Often such search problems can be described by resource constraint evanescent random walkers (searchers) aiming to hit targets containing the resources which they need for their survival. In the first part we consider the target hitting counting process (THCP) of an immortal Markov walker navigating in an ergodic network. We analyze the THCP of an arbitrary stationary set of target nodes. We consider an integer counting variable which counts the hits of the target depending on the departure node. This non-decreasing counting variable is increased by a unit when a target node is hit by the walker. In general, the THCP is not a renewal counting process apart from some distinguished cases. In the second part we connect the THCP with the survival statistics of a mortal walker performing Markov steps in an ergodic network. The survival of the walker requires a positive "budget". Each step reduces the budget by one unit. The budget is reset at target hitting times to an IID copy of its initial value, highlighting the connection with stochastic resetting. The walker dies when the budget reaches null for the first time. We obtain analytically the evanescent propagator matrix, the survival probability of the walker, the mean residence time on a set of nodes during the walker’s lifetime, and the expected lifetime. The results also include the number of target hits (budget renewals) in a walker's lifetime. We identify three pertinent scenarios: (i) the forager scenario, in which frequent encounters with target nodes extend the walker’s lifetime, (ii) a detrimental scenario, where frequent encounters instead reduce it, and a neutral scenario (iii) where the frequency of target node hits has no effect on the lifetime. We corroborate our analytical results with random walk simulations on Barabási–Albert graphs and highly connected target nodes in degree biased walks.