The most common and most likely answer to this crossword clue is the 6 letter word EVENLY
I've seen this crossword clue in The Times
E
V
E
N
L
Y
More clues leading to the result EVENLY
- Archbishop feeding diocese in regular fashion
- Calmly start to live in US city with first lady
- Waugh, perhaps, has name coming up constantly
- See Anglican cleric entering calmly
- With such regularity one appears in lounge
- Name coming up in Waugh, in impartial manner
- Archdeacon enters see regularly
- Cleric visits cathedral city regularly
- Waugh, for instance, gets name raised regularly
- Waugh, say, bringing name forward, maintaining consistency
Buddy explains!

The straight part
We know that "in regular fashion" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in regular fashion = evenly.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EVENLY mean?
EVENLY adverb- So as to form a flat surface.
- In a fair manner of distribution, giving the same amount or number to each; equally.
- In a manner that leaves no remainder.
- In terms of or by means of even numbers.