The most common and most likely answer to this crossword clue is the 4 letter word TEAT
I've seen this crossword clue in The Guardian
T
E
A
T
More clues leading to the result TEAT
- Drink dispenser used regularly in the east
- Half of inked design around middle of her nipple
- Heads off tippler’s excessive alcoholic tendency, attachment to bottle
- Right to withdraw from deal with milk supplier
- Oops, freshly out of potatoes newly dug
- Stimulate a tart’s nipple
- Rosy prospect at last for dummy
- My first drink supplier leaves on time
Buddy explains!

The straight part
We know that "My first drink supplier" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case my first drink supplier = teat.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 TEAT mean?
TEAT noun- The projection of a mammary gland from which, on female mammals, milk is secreted.
- Something resembling a teat, such as a small protuberance or nozzle.
- An artificial nipple used for bottle-feeding infants.