If $C$ is allowed to depend on $\lambda$, just take $C=2\lambda^p$.
If $C$ is not allowed to depend on $\lambda$, take any nonzero $f\in H_{0}^{1}(0,1)$ and let $\lambda\to\infty$. Then the left-hand side of your desired inequality will go to $\int_0^1|f(x)|^2\,dx>0$ whereas its right-hand side will go to $0$, so that your desired inequality will fail to hold.