  We study monodromy reduction of Fuchsian connections from a sheave theoretic
viewpoint, focusing on the case when a singularity of a special connection with
four singularities has been resolved. The main tool of study is {based on} a
bundle modification technique due to Drinfeld and Oblezin. This approach via
invariant spaces and eigenvalue problems allows us not only to explain
Erd\'elyi's classical infinite hypergeometric expansions of solutions to Heun
equations, but also to obtain new expansions not found in his papers. As a
consequence, a geometric proof of Takemura's eigenvalues inclusion theorem is
obtained. Finally, we observe a precise matching between the monodromy
reduction criteria giving those special solutions of Heun equations and that
giving classical solutions of the Painlev\'e VI equation.
