Currently, it's not possible to restart the mysql service(or the underlying VPS) for the managed database offering.
This caused some headaches for me, because my database server was hanging(with deadlock). Usually, this is fixed with simple restart of the database, but when this option is not available, I had to wait for the support to restart the mysql service for me. In this particular case, support was slower than usual, causing more than 1 day of sluggishness for my app.
As you can imagine, this was quite frustrating experience.
AWS RDS has this feature: https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_RebootInstance.html