Arkansas Democrat-Gazette cartoonist John Deering’s political cartoon in the editorial section for April 27 has me puzzled.
I’m not entirely clear as to what the cartoonist’s chief message is, but the secondary message bothers me a lot.
Regard the first two speech balloons:

The first speaker uses the phrase “between me and John Kasich.”
The second speaker seems to be correcting the phrase to “[between] John Kasich and I.”
However, “between me and John Kasich” is grammatically correct, if a little impolite. I would prefer to put the other person first: “between John Kasich and me.”
The phrase “between John Kasich and I” is non-standard English grammar.
Read more about the use of pronouns following prepositions: “Between You and Me” vs “Between You and I.”