Your employer uses your tax code to work out how much tax to take off your wages. If HMRC has sent your employer the wrong tax code, even for just one month in the last 4 years, your employer could easily have taken too much money off you and you’d never know.
HMRC rules make it your responsibility to check your employer has taken the right amount of tax. It’s not your employers job to do this. Your employer must take whatever tax HMRC tells them to take, by using the tax code HMRC sends them. In fact, your employer would probably never know if you were paying too much.