My scredit score is 689, My mothers credit score is 760. If I add my mother as an authorized user to MY credit card, will my score go up?

No it will not go up or down 

No, because it's YOUR credit, not hers, that determines your score. Your payment history, your credit-to-debt ratio, your credit limits--all of those are more important than who is on the card as an authorized user--because THEY are not going to pay those bills, you are. She may reimburse you, but it's YOUR payment profile that makes your score. 

Adding her will do nothing.  It's paying bills on time, and keeping percentage of available credit being used below the trigger points that lower a score, that build a score, along with an account aging.  So it's a combination of wise usage and time that builds a good score.

If you can't pay the balances and need her help paying bills on time, that will keep a score from dropping, and adding her would help there. But it won't increase your score any.

No, but if she added you to hers, it might.  You can increase your score with a credit builder loan from a credit union.

no ...................................

There are so many variables to credit scores that you can't bank on it, but add her anyway.  It can't hurt.

Adding her isn’t going to make your score go up. It doesn’t take the average between the two of you. Your credit score is dependent on your spending habit. 

Your mom can make it go up if she uses it and pays it on time. But it can be lowered if she spends $10 and you spend $1000.  The minimum payment is $20. So your mom pays her $10 but you don’t pay your $10 to make the minimum payment, it’ll go down. On the flip side, if she spends $1000 and you spend $10, then it can go up if she pays her 1000 and you don’t pay your 10. Sometimes it’s better to carry a debt than to carry nothing. 

Don't know about your credit score but your mother's credit score would go down.  She'd be foolish to hook her credit up with yours.  Leave her out of your financial situation.