He, at any rate, supposed he would speak with them.
Here's why:
* "He" is the subject of the sentence, and needs to be capitalized.
* "At any rate" is a parenthetical phrase, meaning it can be removed from the sentence without changing the meaning. Parenthetical phrases are set off with commas.
* "Supposed" needs to be followed by a verb, in this case, "he would speak".
* "Them" is the object of the verb "speak".
The sentence is now grammatically correct and easy to read.