We introduce an accurate and efficient method for a class of nonlocal potential evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipolar potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel and Taylor expansion of the density. Starting from the convolution formulation, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. Hence, the potential is separated into a regular integral and a near-field singular correction integral, where the first integral is computed with the Fourier pseudospectral method and the latter singular one can be well resolved utilizing a low-order Taylor expansion of the density. Both evaluations can be accelerated by fast Fourier transforms (FFT). The new method is accurate (14-16 digits), efficient ($O(N \log N)$ complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelable.