Plug $x=2$ and $h(x)=6$ into $h(x)=k(0.5)^x+4$ to create an equation with one unknown, $k$.

After substituting, simplify $(0.5)^2$, then isolate $k$ by subtracting 4 from both sides and dividing by the coefficient in front of $k$. Once you have $k$, rewrite $h(x)$ and look for that equation among the choices.

In Desmos, type h(x)=k*(0.5)^x+4. When prompted, make k a slider so you can adjust its value.

Add a point by typing (2,6); it will appear on the graph. Adjust the slider for k until the curve of $h(x)$ passes exactly through the point $(2,6)$. Note the value of k at that moment.

Once you know the value of k from Desmos, rewrite the function in the form $h(x)=k(0.5)^x+4$, then select the answer choice whose coefficient on $(0.5)^x$ and constant term match your function.

We are told that h has the form $h(x)=k(0.5)^x+4$, where $k$ is a constant. This means the last term must be $+4$ in the correct answer, and only the coefficient $k$ in front of $(0.5)^x$ can change.

The point $(2,6)$ lies on the graph of $h$, which means when $x=2$, $h(x)=6$.

So plug $x=2$ and $h(x)=6$ into $h(x)=k(0.5)^x+4$:

First simplify $(0.5)^2$:

So the equation becomes

Subtract 4 from both sides:

Now divide both sides by $0.25$ to solve for $k$:

Compute $\dfrac{2}{0.25}=8$, so $k=8$.

Substitute back into the general form $h(x)=k(0.5)^x+4$ to get

Comparing with the choices, this matches option C.