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Choice C is correct. It's given that the equation $-9x^2+30x+c=0$ has exactly one solution. A quadratic equation of the form $a{x}^2+bx+c=0$ has exactly one solution if and only if its discriminant, $-4ac+b^2$, is equal to zero. It follows that for the given equation, $a=-9$ and $b=30$. Substituting $-9$ for $a$ and $30$ for $b$ into $b^2-4ac$  yields ${30}^2-4\left(-9\right)\left(c\right)$, or $900+36c$. Since the discriminant must equal zero, $900+36c=0$. Subtracting $36c$ from both sides of this equation yields $900=-36c$. Dividing each side of this equation by $-36$ yields $-25=c$. Therefore, the value of $c$ is $-25$. 

Choice A is incorrect. If the value of $c$ is $3$, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

Choice B is incorrect. If the value of $c$ is $0$, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

Choice D is incorrect. If the value of $c$ is $-53$, this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.