Would a half aunt share x-dna with her father's granddaughter? The situation is simple, but we are missing a link somewhere. The two match at 1160cM, however one is in her 60's, the other in her 30's, so the most likely connection is either Half aunt or 1st cousin. In gedmatch 1 to 1 comparison there is no x dna shared by the two and due to other matches already established we know this is on the elder's father's side. So we're thinking the younger is a half siblings child placed for adoption.
I assume that you're talking about the daughter of a half-sister here. Half-sisters with a father in common will inherit the same X-chromosome in its entirety. A biological daughter of either of them will inherit some recombination of her mother's 2 X-chromosomes, so there should be significant x-DNA in common between the daughter and each of the half-sisters, but not the same match. The likelihood of a daughter sharing no x-DNA with her mother's half-sister is microscopically small.
If, on the other hand, you're talking about a half-brother and -sister with a common father, then the half-brother's daughter would be expected to share no x-DNA with her half-aunt, since her father inherited only y-DNA from the common ancestor.