Update values from a column of one table if they exist in another table's column if these values are unique

I have a column filled with values that need updating because they are too ambiguous and I want to update these values with values from another table columns.

Player Table:

Sports Table:

In this example I am trying to replace values in the Players column with the player's Email because some players have the same first name. I don't believe that there is a way to find out which Bob and Alex are in the Sports Table so I want to do the minimum and change the column values for the names do not have duplicate first names. The updated Sports Table should look like below.

Updated Sports Table:

My first thought was to find all column values in the Player table that did not have any First Names that showed up more than once and I managed to get that far with the following query below.

SELECT Email, FirstName
FROM PlayerTable
GROUP BY Email, FirstName
HAVING COUNT(P.FirstName) = 1

However, I haven't been able to find a way to use this to update the Sports Table correctly. I tried the following lines of code, but that just outputted an incorrect table.

UPDATE SportsTable
SET Player = (
    SELECT P.Email, P.FirstName
    FROM PlayerTable P
    WHERE P.FirstName = Player
    GROUP BY P.Email, P.FirstName
    HAVING COUNT(P.FirstName) = 1
)

You are basically on the right track. This query will return all First Names and their email that appear with one email only:

SELECT 
  FirstName,
  MIN(Email) AS Email
FROM Player
GROUP BY 
  FirstName
HAVING 
  COUNT(DISTINCT Email) = 1;

So you need to check the distinct count of emails per FirstName rather than count the FirstName itself.

You can use above query in a CTE or in a pure subquery and join it to the Sports table. I think both ways are fine.

Here the option with a CTE:

WITH uniqueMails AS
(SELECT 
  FirstName,
  MIN(Email) AS Email
FROM Player
GROUP BY 
  FirstName
HAVING 
  COUNT(DISTINCT Email) = 1)
UPDATE s
SET s.Players = u.Email
FROM Sport s
JOIN uniqueMails u
ON s.Players = u.FirstName;

Verify it works as expected on this db<>fiddle with your sample data.

And here the option without CTE:

UPDATE s
SET s.Players = uniqueMails.Email
FROM Sport s
JOIN 
(SELECT 
  FirstName,
  MIN(Email) AS Email
FROM Player
GROUP BY 
  FirstName
HAVING 
  COUNT(DISTINCT Email) = 1) uniqueMails
ON s.Players = uniqueMails.FirstName;

db<>fiddle for this option.