Can a MySQL trigger include a where clause
You're almost there. Your trigger should probably look like this:

CREATE TRIGGER update_sire_name before UPDATE ON table_name
for each row

    set new.sireName = 
    (select name from table_name where regNo = new.sireRegNo)


dbfiddle



1 Yes, there can be a where clause in a trigger.

2 Consider using numeric data types (e.g. INTEGER) where values are clearly numeric (e.g. regNo)