  A well studied classical problem is the harmonicity of functions satisfying
the restricted mean-value property (RMVP) for domains in $\mathbb{R}^n$.
Recently, the author along with Biswas investigated the problem in the general
setting of Riemannian manifolds and obtained results in terms of unrestricted
boundary limits of the function on a full measure subset of the boundary.
However in the context of classical Fatou-Littlewood type theorems for the
boundary behavior of harmonic functions, a genuine query is to replace the
condition on unrestricted boundary limits with the more natural notion of
non-tangential boundary limits. The aim of this article is to answer this
question in the local setup for pre-compact domains with smooth boundary in
Riemannian manifolds and in the global setup for non-positively curved Harmonic
manifolds of purely exponential volume growth. This extends a classical result
of Fenton for the unit disk in $\mathbb{R}^2$.
