The X Factor star was due to perform at the Scarborough Open Air Theatre - but the adverse weather forced organisers to consider rescheduling.
A statement from Cuffe and Taylor said: "Due to the adverse weather conditions, and the continued lightning strikes in the Scarborough area, this evening's performance with James Arthur is unable to proceed.
"We extend our sincerest apologies to everyone who has come to the show but safety must come first.
"We are investigating the possibility of rescheduling the show and will post an update as soon as possible."
The weather comes as the British heatwave brought the hottest day of the year so far, with temperatures soaring to 35C.
Arthur wrote on Twitter: "Devastatingly due to the thunder and lightning in Scarborough this evening the promoters and the council have decided it's not safe for the show to go ahead.
"The gig will be rescheduled and if you want you can get a refund. I am so gutted and so sorry to all who came out."
Replying to a fan who said it was not raining in the area anymore, he said: "I think it's just precautionary, good chance it will continue throughout the night.. dangerous to do a show with all the lights etc with lightning."
Arthur is scheduled to play at Outside Live in Norfolk on Friday.