In Whitby there are apparently music sessions every night, I am informed.

Sunday and Monday, the Station, singing / acoustic guitars, not particularly folk based.
Tuesday, Elsinore, singing / tunes session, more folk based.
Wednesday, Friday, the Dolphin.
Thursday, folk club, I think at the Friendship Rowing Club
Saturday, First In , Last Out.   

No doubt you'll find much more accurate information during Folk Week!

Hope you find what you want,

Joy