The guest is not blackmailing her/him per se. The guest had a bad experience on the host's property due to the construction in the area, the guest demanded a refund, got denied, and the guest came back to the property since the guest did pay for the stay.
If I was in that situation, I would have accommodated the guest in another property with pro rated refund and if there would have been a difference, I would have absorbed it or negotiated it somehow; get creative. Furthermore, I would have gone to the city, apartment management, and the construction company and asked/demanded why there was no notification regarding the nearby construction. All permitted work requires advanced notice pre construction. Whoever failed to do their job correctly is the one that you send the bill to (except for city - they never pay).
As hosts, we live off of our reviews and we have to protect our reviews at all costs. I had a gentlemen who confirmed he would arrive one day early and if I could hold the date for him. I forgot all about it and on the day of arrival, I (key word "i") searched for a nearby property, provided him the link and told him to book it. I advised that since he would have paid X amount for the night on my property, I would pay the difference. So I paid the total for one night minus what my property would have cost him.
At the end of the day, the guest did not review my property BUT, the outcome would have been different if I left him out in the cold.
In the OP's case, the guest has all rights to be angry given, if in fact, there was construction going on around the property.
It was an unfortunate event for both parties.
C. Brothers Property Management, llc