Wiktionary
tinker's damn
n. An insignificant or worthless amount.
WordNet
Usage examples of "tinker's damn".
It had been Baird who had persuaded his two colleagues to act, or rather he had persuaded General Harris to act much against that officer's better judgement, and Baird frankly did not give a tinker's damn whether Colonel Arthur bloody Wellesley approved or not.
If they wanted us for anything they'd come and get us, but they wouldn't give a cockeyed tinker's damn for our money.
You go to a shop and buy a thing and the shopkeeper don't give a tinker's damn if you pay or not.