Let $(M,g)$ be a complete Riemannian manifold, let $p \in M$, and define the distance function $r_p: M \to \mathbb{R}$ by $r_p(x) := d(p,x)$. Let $q \in M \setminus \{p\}$, set $R := r_p(q)$, and let $\gamma: [0,R] \to M$ be a unit-speed minimizing geodesic from $p$ to $q$, so that $\gamma(0)=p$ and $\gamma(R)=q$.
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For each $\delta \in (0,R)$, define $p_\delta := \gamma(\delta)$ and define
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\begin{align*}
r_\delta: M &\to \mathbb{R} \\
x &\mapsto \delta + d(p_\delta,x).
\end{align*}
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Then $r_\delta(x) \ge r_p(x)$ for every $x \in M$, and $r_\delta(q)=r_p(q)$. Moreover, if $q$ is not in the cut locus of $p_\delta$, then there exists an open neighbourhood $U_\delta \subset M$ of $q$ such that $r_\delta|_{U_\delta}$ is smooth and is an upper barrier for $r_p$ at $q$, meaning
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\begin{align*}
r_\delta(x) \ge r_p(x) \quad \text{for all } x \in U_\delta,
\qquad
r_\delta(q)=r_p(q).
\end{align*}