Removing Brackets and the content
I've set up this example:

create table tbl (horse_name varchar(100));
insert into tbl values ('Rocket(FRA)'),('Apple Jade(IRE) 2'),('Zac');


If last parenthese is located always at end of the string you could use next syntax:

select case when locate('(', horse_name) = 0 
            then horse_name
            else left(horse_name, locate('(', horse_name) - 1)
       end as horse_name            
from   tbl;


But if there are additional characters after the last parenthese you could CONCAT left and right substrings in this way:

select case when locate('(', horse_name) = 0 
            then horse_name
            else concat(left(horse_name, locate('(', horse_name) - 1),
                        right(horse_name, length(horse_name) - locate(')', horse_name)))
       end as horse_name            
from   tbl;


This is the final result (note I've added a 2 after (IRE):



| horse_name   |
| :----------- |
| Rocket       |
| Apple Jade 2 |
| Zac          |



db&lt;>fiddle here