As Jason said, you can't really direct the cultural expansion. It is not exactly random however. I haven't looked directly at the algorithm, but from my experience
- It will prioritize tiles which have resources over those that don't
- It will prioritize workable tiles over those that aren't (out of range, mountains/snow, etc)
- It will prioritize tiles with higher food/production/etc. values over lower.
You can't directly snag that tile, but if you have enough spare gold, you can influence it to try and select that tile for cultural expansion, essentially by buying up the ones it would select.
In your case, if you bought up the 2 remaining deer and the salt, you probably have a decent shot at the gold getting selected. Your city is small enough that it might decide you have enough workable land headroom that it'll go for the resource. You might have to go as far as buying up all the other tiles, barring the tundra and maybe the marsh.
Failing that, grabbing at least the existing resources would both give your city a decent start at growth, and give you a head start to the culture growth finally reaching it. It doesn't look like there's a rival (at least not in the field of that screenshot) so you might have time, if you're not hurting for the happiness or a city-state quest.