In this paper, we study the optimum sensing of a time-varying random event with a sensor powered by energy harvesting devices. The system aims at reconstructing a band-unlimited continuous-time random process by using discrete-time samples collected by a sensor. Due to the random nature of the harvested energy, the sensor might not have sufficient energy to perform a sensing operation at a desired time instant. We propose a best- effort Poisson sensing policy. The sensing policy defines a set of random candidate sampling instants following a Poisson point process (PPP) in the time domain. At a given candidate sampling instant, the sensor collects a sample if there is sufficient energy to do so, and does nothing otherwise. By analyzing the statistical properties of the best-effort Poisson sensing policy, we develop an optimum estimator of the underlying random event. The asymptotic mean-squared error (MSE) of the estimation are expressed as an explicit closed-form expression of several key system parameters, such as the ratio between the average energy harvesting rate and consumption rate, the time correlation of the random event of interests, and the energy allocation between sensing and transmission. %The analytical results are used to identify optimum system operation parameters. Numerical results show that the proposed best-effort Poisson sensing policy outperforms existing uniform sensing policies.