You detectives have been busy (and done well!)
Yes, this is the right newspaper report and right family.
I have a copies of maps of the Beauchief area drawn in around 1811 (not available on line, and I can't scan them as they're not great copies) and Great Tom Cross field was to the north of the eastern end of Abbey Lane - I'm pretty sure that it was close to what's now Periwood Lane, and Abbey Lane cemetery now covers that area. It was one of several fields owned by Benjamin Steade, occupier of Beauchief Hall, and farmed by Martha's father. I don't know where the Hawksleys lived but it would have been local to the fields, maybe in the Woodseats area; the children were baptised and buried at Beauchief Abbey, just up the road from there.
So that's why I expect Bagshaw Wood to be roughly to the north of Abbey Lane, between there and Sheffield.
I think the Joseph Howksley, farmer of Owlerton, may have been a relative but I don't think it was Martha's father. Again, it's a fair distance to travel from there to Abbey Lane (a 5-6 mile walk each way). There were quite a few Dysons around and I have tried to find where Harriet and her husband lived, but not been successful. On their marriage licence, Thomas was of Ecclesall-(nb not Ecclesfield!) but that was 15 years before this incident so clearly they could have moved.
Edited - just checked my records and I'd forgotten that Joseph had left a will, which mentioned his dwelling house on Abbey Lane; this was in 1836.