Fishy letter to a Shakespearian shrew (5)

I believe the answer is:

skate

I'm a little stuck... Click here to teach me more about this clue!

'fishy' is the definition.
(type of fish)

'shakespearian shrew' is the wordplay.
'shakespearian' becomes 's' (this could be a standard abbreviation I've not previously seen).
'shrew' becomes 'kate' (Katherina or Kate in The Taming of the Shrew).
's'+'kate'='SKATE'

'letter to a' is the link.
This may not be right. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for skate that I've seen before include "Runner" , "Large, edible rays" , "Complete, 19-less" , "Cartilaginous fish" , "Type of ray" .)

I've seen this clue in the Evening Standard.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.