Конечно, давайте разберём, какие глагольные формы нужно использовать в данном письме:
do - Since you want to know what I am doing next week, I thought I’d write and let you know. (Present Continuous, поскольку описывается действие, запланированное на будущее время.)
be - It is going to be a very busy week. (Present Simple, так как утверждение о будущем используется в общем смысле.)
go - On Monday I am going to York. (Present Continuous, так как это запланированное действие в ближайшем будущем.)
probably / be - I will probably be there for three days, and by Wednesday I. (Future Simple с наречием "probably" для выражения вероятности.)
meet - .and by Wednesday I will have met every important artist in the town. (Future Perfect, чтобы показать завершённое действие к определённому моменту в будущем.)
go - If everything goes well, I will go to Newcastle on Thursday morning. (Future Simple, так как это действие, зависящее от условия.)
meet - There I will meet the chairman of the Arts Council. (Future Simple, для выражения будущего действия.)
visit - Then on Friday and Saturday I am going to visit several small towns in the area to see what their galleries are like. (Be going to, так как это намерение или план.)
travel - By Sunday I will have been traveling for days and I imagine I. (Future Perfect Continuous, для выражения продолжительности действия до определённого момента в будущем.)
be - .I imagine I will be very tired, so it looks like I. (Future Simple, для выражения будущего состояния.)
not / come - .so it looks like I will not come to your party on Sunday night. (Future Simple в отрицательной форме, для выражения будущего действия.)
invite - I hope you will invite me to the next one. (Future Simple, для выражения надежды на будущее действие.)
Полный текст с правильными формами глаголов:
Dear Debbie,
Since you want to know what I am doing next week, I thought I’d write and let you know. It is going to be a very busy week. On Monday I am going to York. I will probably be there for three days, and by Wednesday I will have met every important artist in the town. If everything goes well, I will go to Newcastle on Thursday morning. There I will meet the chairman of the Arts Council. Then on Friday and Saturday I am going to visit several small towns in the area to see what their galleries are like. By Sunday I will have been traveling for days and I imagine I will be very tired, so it looks like I will not come to your party on Sunday night. Sorry! I hope you will invite me to the next one.
Give my love to Mike.
Love,
Susan