How can I check if $n >m$ in $a^n b a^m$?
Start with placing tape over $b$, then go to the rightmost $a$, put blank, go back to $b$ and delete the rightmost $a$.
Every time you scratch out $a$s on both sides, $n - m$ does not change until one side is blank.
Check if the left side is not empty, delete $b$ (and if it is required, go back to the beginning of the tape) and accept, otherwise reject.