  The purpose of this work is to introduce a strategy for determining the nodes
and weights of a low-cardinality positive cubature formula nearly exact for
polynomials of a given degree over spherical polygons. In the numerical section
we report the results about numerical cubature over a spherical polygon $\cal
P$ approximating Australia and reconstruction of functions over such $\cal P$,
also affected by perturbations, via hyperinterpolation and some of its
variants. The open-source Matlab software used in the numerical tests is
available at the author's homepage.
