I am a landlord selling my property. To put it on sale I had to have the tenants leave. To get them to leave early I had to pay them one month’s free rent. Can I deduct that rent concession in th

If by "I had to pay them one month's free rent", you actually mean that you did not charge them rent for their last month there, then the answer to your question is no. You can not deduct from your income that which you never actually received and paid taxes on in the first place.

I gave them one month’s rent in cash. So cash went from my hand to theirs.

since you actually paid them out of pocket it is deductible. 

If you claimed that money as income and reported it as same, then you can claim giving it back to them as a concession. That basically makes it a wash on the tax return.