Has\Have

[RoseSpring]

The minister, as well as the guards, has\ have gone for a jog.

[Rover_KE]

The minister, as well as the guards, has gone for a jog.

'...as well as the guards' can be omitted - leaving 'The minister has gone for a jog'.

Alternatively, you can say 'The minister and the guards have gone for a jog'.

Rover

[philadelphia]

*Not a teacher

The other way around would be: 'Neither the minister nor the guards HAVE gone for a jog'.