Realistic modeling of energy consumption is crucial for accurate performance analysis of wireless-powered sensor nodes. In this paper, we analyze the performance of wireless-powered sensor transmissions taking into account both the energy cost of sensing and transmission. We consider a sensor that is harvesting energy from an ambient radio-frequency (RF) signal and using this energy to perform sensing and transmission. Since energy harvesting is time-varying in nature, it introduces a delay in the sensor transmissions. We study two delay-related metrics, one measuring how frequent the sensed information is updated at the sink and the other measuring the time taken from the sensing operation to successful transmission of sensed information. We analytically characterize the statistical behavior of both metrics and find an important tradeoff between them. In particular, our results illustrate that more frequent update of sensed information at the sink increases the time taken from the sensing operation to successful transmission of sensed information.