  Deng (arXiv:math/9812082) gave an asymptotic formula for the number of
rational points on a weighted projective space over a number field with respect
to a certain height function. We prove a generalization of Deng's result
involving a morphism between weighted projective spaces, allowing us to count
rational points whose image under this morphism has bounded height. This method
provides a more general and simpler proof for a result of the first-named
author and Najman on counting elliptic curves with prescribed level structures
over number fields. We further include some examples of applications to modular
curves.
