he asked his mother that whether she knew what was going on then的错误是

he asked his mother that whether she knew what was going on then的错误是

多一个that
改成：He asked his mother whether she knew what was going on then.
whether she knew what was going on then是个宾语从句,作谓语动词ask的宾语（直接宾语,表示问话内容）,whether本身可以引导宾语从句,不需要重复使用关联词了

不用中间的that
he asked his mother whether she knew what was going on then