So I’m guessing you want all results in the Washing & Laundry category where Washing machine is in the name field. If this is the case you can simply add the AND keyword to narrow it down further
$query="SELECT * FROM shop WHERE mcat='Washing & Laundry' AND name='washing machine'";
take care of the case sensitivity of the value in the field “name”. If it defined something like latin1_swedish_ci, means “case-insensitive” then no problem
otherwise you will have to handle the collation in your query/db.