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Using a perturbative technique, in this work we study the deflection of null and timelike signals in the extended Einstein-Maxwell spacetime, the Born-Infeld gravity and the charged Ellis-Bronnikov (CEB) spacetime in the weak field limit. The deflection angles are found to take a (quasi-)series form of the impact parameter, and automatically takes into account the finite distance effect of the source and observer. The method is also applied to find the deflections in CEB spacetime with arbitrary dimension. It's shown that to the leading non-trivial order, the deflection in some $n$-dimensional spacetimes is of the order $\mathcal{O}(M/b)^{n-3}$. We then extended the method to spacetimes that are asymptotically non-flat and studied the deflection in a nonlinear electrodynamical scalar theory. The deflection angle in such asymptotically non-flat spacetimes at the trivial order is found to be not $π$ anymore. In all these cases, the perturbative deflection angles are shown to agree with numerical results extremely well. The effects of some nontrivial spacetime parameters as well as the signal velocity on the deflection angles are analyzed.