In this case the character is “^“.
For example in the following format with 3 columns.
From:
1 cccc cccc
2 cta^<t cta^<tcc
3 ccc cccc^*
4 ^]a^*c^*c ^]a^*c^*c
To:
1 cccc cccc
2 ctat ctatcc
3 ccc cccc
4 acc acc
I’m quite sure this is possible with sed.
To match your demonstration input/output, it should just be
s/\^.//g. To match the description, you needs/\^./\^/g(i.e., you’ve described it as removing the following character, but shown removing both the caret and the following character).