The minister, as well as the guards, has\ have gone for a jog.
Retired English Teacher
'...as well as the guards' can be omitted - leaving 'The minister has gone for a jog'.
Originally Posted by RoseSpring
Alternatively, you can say 'The minister and the guards have gone for a jog'.
Student or Learner
*Not a teacher
The other way around would be: 'Neither the minister nor the guards HAVE gone for a jog'.