I'm sorry, but I have something I have to do on this day, so I won't be able to make it to our lesson.
I'm afraid I have to do something on this day, so I can't be there for our piano lesson.
ご質問ありがとうございます。
① "I'm sorry, but I have something I have to do on this day,"＝「すみません、でもこの日にはしなければいけないことがあり」
"so I won't be able to make it to our lesson."＝「レッスンには出られません。」
② "I'm afraid I have to do something on this day,"＝「残念ながらこの日はしなければいけない案件があり」
"so I can't be there for our piano lesson."＝「ピアノのレッスンには出られません。」
ご参考までに。
Sorry, I can't make it to the lesson that day. I have some errands to run.
I won't be able to attend our lesson that day. I have something I need to do.
この場合、次のような言い方ができますよ。
ーSorry, I can't make it to the lesson that day. I have some errands to run.
「すみません、その日は用事があってレッスンを都合がつけられません」
to make it で「都合をつける」
to have some errands to run で「用事がある」
ーI won't be able to attend our lesson that day. I have something I need to do.
「その日は用事があってレッスンに出れません」
to attend one's lesson/class で「レッスン/授業に出る」
to have something I need to do「やらなければいけないことがある」＝「用事がある」
ご参考まで！
アメリカ合衆国 
カナダ 
