Can we go to another galaxy, another continuum, etc. using the mycelial network?
No. S4 directly makes it a point that they can only travel to the edge of the Milky Way galaxy. From there, they have to use warp (making it through the edge of the galaxy notwithstanding). From The Galactic Barrier (DIS S4E10)
NILSSON: The mycelial network thins as it approaches the galactic barrier, so Commander Stamets expects us to arrive about four light-years away. We'll warp from there.
Alternate universe mycelial network
So... how did Discovery go to the Mirror Universe using the spore drive? There's some key things to consider (lots of spoilers)
- Lorca had been working on the Spore Drive project for some time. He knew the system very well by this point
- Lorca had an elaborate plan to make the drive work the way he wanted. When Burnham is piecing Lorca's plan together, she is reminded of the 133 jumps Stamets made in S1E9 "Into the Forest I Go".
- Lorca was from the Mirror Universe and apparently had enough knowledge to get back to where he had already been.
Tarka had never been to Kayalise and his research was centered around transporters, not the Spore Drive. The only people who knew that the Spore Drive could even do something like that were on Discovery, and they had good reason not to talk about it. In short, it's not impossible it could do that, but Tarka had already sunk all his efforts into powering the transporter effort (including betraying lots of people to get there).