  The main result of this paper supports a conjecture by C. P\'erez and E. Rela
about a very recent result of theirs on self-improving theory. Also, we extend
the conclusions of their theorem to the range $p<1$.
  As an application of our result, we give a unified vision of weighted
improved Poincar\'e-type inequalities in the Euclidean setting, which gathers
both weighted improved classical and fractional Poincar\'e inequalities within
an approach which avoids any representation formula. We improve some already
known results.
  Finally, we also explore analog inequalities in the context of metric spaces
by means of the already known self-improving results.
