Step 1: Define what makes a wave progressive.
A progressive (travelling) wave is one where the waveform moves through space. Mathematically, the displacement must be a function of $(x - vt)$ or $(x + vt)$ for some speed $v$.
In sinusoidal form, this looks like $A\sin(kx - \omega t)$ or $A\cos(kx + \omega t)$. The key feature is that $x$ and $t$ are combined as $(kx \pm \omega t)$, not kept separate.
Step 2: Test option (A): $y = 2\cos 3x \sin 10t$.
Here $x$ and $t$ appear in separate multiplied factors. Using the product-to-sum identity, this equals $\sin(10t + 3x) - \sin(10t - 3x)$, which is a superposition of two waves travelling in opposite directions. That describes a standing wave, not a progressive wave.
Step 3: Test option (B): $y = 2\sqrt{x - vt}$.
Although $(x - vt)$ appears, the square root means the function is undefined (imaginary) when $x < vt$. A valid wave function must be defined for all positions at all times. This is not a physically valid wave.
Step 4: Test option (C): $y = 3\sin(5x - 0.5t) + 4\cos(x - 0.5t)$.
Both terms are of the form $f(kx - \omega t)$. Each individual term is a progressive wave moving in the $+x$ direction. A superposition of progressive waves (with the same sign convention) is also a progressive wave. This option qualifies.
Step 5: Test option (D): $y = \cos x \sin t + \cos 2x \sin 2t$.
Both terms have $x$ and $t$ separated by multiplication. Each term is a standing wave pattern. Their sum is also a standing wave arrangement, not a progressive wave.
Step 6: State the answer.
Only option (C) represents a progressive wave.
\[
\boxed{\text{C}}
\]