The most common and most likely answer to this crossword clue is the 7 letter word PRIMATE
I've seen this crossword clue in The Times
P
R
I
M
A
T
E
More clues leading to the result PRIMATE
- Archbishop in particular put away
- High churchman's strait-laced spouse missing mass
- Ape an archbishop?
- Ape a man of the cloth
- Archbishop is formal at start of evensong
- Formal note about a church dignitary
- Who’s in charge of bishops? Big ape, perhaps
- Archbishop’s affectedly proper answer leads to Times’ editorial
- Lip bitten by head clergyman
- Thomas Becket? Proper troublemaker!
Buddy explains!

The straight part
We know that "church dignitary" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case church dignitary = primate.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 PRIMATE mean?
PRIMATE noun- A mammal of the order Primates, including simians and prosimians.
- A simian anthropoid; an ape, human or monkey.
- In the Catholic Church, a rare title conferred to or claimed by the sees of certain archbishops, or the highest-ranking bishop of a present or historical, usually political circumscription.
- In the Orthodox Church, the presiding bishop of an ecclesiastical jurisdiction or region. Usually, the expression primate refers to the first hierarch of an autocephalous or autonomous Orthodox church. Less often, it is used to refer to the ruling bishop of an archdiocese or diocese.
- In the Anglican Church, an archbishop, or the highest-ranking bishop of an ecclesiastic province.