10. 8 .

The point is $F\left(\frac{1}{2}, 0\right)$. The equation of the semicircle is $\left(x-\frac{9}{2}\right)^{2}+y^{2}=16(y \geq 0)$. Substituting $y^{2}=2 x$ into the semicircle equation yields $x^{2}-7 x+\frac{17}{4}=0$. Let $M\left(x_{1}, y_{1}\right), N\left(x_{2}, y_{2}\right)$, then by the definition of the parabola and the given theorem, $|M F|+$ $N F \left\lvert\,=x_{1}+x_{2}+2 \times \frac{1}{2}=8\right.$.
Let the focus of the parabola $y^{2}=2 x$ be $F$. A circle is drawn with $P\left(\frac{9}{2}, 0\right)$ as the center and $|P F|$ as the radius, intersecting the parabola above the $x$-axis at points $M$ and $N$. Then the value of $|M F|+|N F|$ is