My child will be really disappointed to know that she can't take your lesson next week.
My child will be sad to learn that he won't be able to take lessons from you tomorrow.
ご質問ありがとうございます。
① "My child will be really disappointed to know that"＝「私の子は～と知ってとてもがっかりするでしょう」
"that she can't take your lesson next week."＝「あなたのレッスンを来週受けることが出来ないと（知って）。」
② "My child will be sad to learn that"＝「私の子は～だと知って、悲しがる（寂しがる）でしょう」
"he won't be able to take lessons from you tomorrow."＝「明日、あなたのレッスンを受けることが出来ない（と知って）。」
ご参考に。
My child will be sad to know that he (she) can't take your class next week.
y child will be disappointed to know that he (she) can't take your class next week.
ご質問ありがとうございます。
「あなたのレッスンができないと分かったら子供が寂しがります」は英語で「My child will be sad to know that he (she) can't take your class next week.」と言います。
「sad」の代わりに「disappointed」でもよく言いますので、「My child will be disappointed to know that he (she) can't take your class next week.」でも言えます。
ご参考になれば幸いです。
アメリカ合衆国 
