A straightforward discretization of problems in d spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives. We propose a nonequispaced hyperbolic cross FFT based on one hyperbolic cross FFT and a dedicated interpolation by splines on sparse grids. Analogously to the nonequispaced FFT for trigonometric polynomials with Fourier coefficients supported on the full grid, this allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes.