Expected Utility and Jensen's Inequality
Didn't manage to get to a definitve answer in one shot, but it seems to me that Jensen inequality is not going to help much.

Build up: 

You are essentially asking that 

\begin{equation}
   E_v \left(u(a - v) \right) \leq E_c \left(u(a-c) |a\leq c \leq b\right)
\end{equation}

for a function $u$ increasing and concave and $[a, b] \subset Supp_v = [x,y]$.

In the language of measures, the above condition can be rewritten as

\begin{equation}
   \int_x^a u(a-c) dG(c) + \int_b^y u(a-c)dG(c) \leq \int_a^b u(a-c) \cdot \left[\frac{f(c)}{F(b)-F(a)} - g(c) \right]dc.
\end{equation} 

First term on the left is certainly positive, second term on the left is certainly negative. The term on the right is of undetermined sign: $u(a-c)$ is negative in the integration interval, but $\frac{f(c)}{F(b)-F(a)}$ cannot be uniformly lower than $g(c)$ on $[a,b]$, as the former on this interval must sum up to 1 and the latter to $G(b)-G(a) \in (0,1)$.