Exploiting proper controller and bearing-only measurement to perform the target tracking with unmanned aerial vehicles (UAV) has become a popular trend due to the convenience of single bearing sensor and the flexibility of UAV. This article revisits the slowly moving target circumnavigation problem with UAV modelled as first-order integrator, presenting results in three aspects. First, we focus on the problem with discrete bearing data and control input, illustrating the principle parameter conditions for system convergence. Second, the thorough convergence analysis from a different perspective for a typical algorithm is presented, resulting the more accurate conclusion upon the target-circumnavigation system convergence. Third, based on the estimation property, an algorithm that exploits the rough information of target velocity is proposed, which significantly accelerates the estimation error convergence rate and improves the estimation performance. Simulation results are given that validates the conclusion that the estimation error and UAV-distance can be unbounded when the target velocity is specially designed under common assumptions in the literature, which corrects the conclusion in existing work, and also validates that the proposed algorithm possesses estimation performance obviously superior to that of existing algorithms especially for situation with constant-velocity target.