Apologies -- it is indeed 'flyspell-auto-correct-previous-word' (and its 
keybinding 'C-;') that I intended to write! However, this function 
*does* cycle through candidates when invoked repeatedly in emacs 26.0.91 
and does *not* do so in the 26 Rc-1. You are right that the function 
does not explicitly state that there will be this cycling behavior, but 
it is at least a change between emacs versions, and a negative one (it 
seems to me anyway) at that.

Is there other information I could give you (sorry - this is my first 
"bug" report).

__

Colin McLear
colinmclear.net

On 17 Apr 2018, at 10:52, Eli Zaretskii wrote:

>> From: "Colin McLear" <mclear@fastmail.com>
>> Date: Mon, 16 Apr 2018 22:45:46 -0500
>>
>> 'flyspell-auto-correct-word' doesn't seem to be working correctly.
>> According to the manual "If invoked repeatedly on the same position, 
>> it
>> cycles through the possible corrections of the current word." 
>> However,
>> it doesn't respond to repeated invocations. It just changes the 
>> spelling
>> to the first "correct" word it finds. Recipe for base emacs below.
>>
>> Starting from emacs -Q, load flyspell in scratch buffer. Spell a word
>> incorrectly. Use 'C-;' to run 'flyspell-auto-correct-word'. Expected
>> result is that you cycle through some different word options until 
>> you
>> get the one you want. Actual result is that it changes the spelling 
>> to
>> the first option and doesn't cycle through any further. I don't have
>> this problem on the immediately prior developer release of emacs 26
>> (i.e. emacs 26.0.91).
>
> I cannot reproduce this if I invoke flyspell-auto-correct-word.
> However, C-; is not bound to flyspell-auto-correct-word, it's bound to
> flyspell-auto-correct-previous-word, whose documentation doesn't
> promise to cycle through candidate corrections, and which is not
> mentioned in the Emacs manual.
>
> Could it be that you intended to type C-. or ESC TAB instead?