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While you can try spot treating it with things like boiling water, vinegar, or Grass-B-Gon, it will also kill your ground cover if it touches it. One way to avoid this is by using some type of shield to cover the plants you wish to keep. Since this is not an easy task, you may need to resort to hand pulling the grass instead.

If they are getting leggy there, that means that the shade is too deep. You can try trimming it down to reinvigorate, but eventually it will get leggy again. Is there anyway you can thin out the canopy over it? Even a little bit of judicial thinning of the branches overhead will allow a little more light in without disturbing the shaded feel of the location. With just a little more light, the plants should be fine there after trimming.

Controlling Pachysandra, this is the question!
Keeping an eye on the plant and keeping it controlled can be an issue.
It's roots are invasive and plastic edging may slow it down but it will find it's way around, under or over it.
Here are a few links.

https://www.gardeningknowhow.com/ornamental/groundcover/pachysandra/pachysandra-ground-cover.htm
https://www.gardeningknowhow.com/special/shade/ground-covers-for-shade.htm
https://www.gardeningknowhow.com/ornamental/groundcover/pachysandra/pachysandra-weed-control.htm

Really the only way to control goutweed (aka: bishop's weed) is to hand pull it and mulch around the area where it keep popping up. You could try using Roundup, or even boiling water, and spot treat the goutweed while covering your pachysandra with a shield of some kind (like cardboard). Still the best way would be to simply pull or dig it up.

You can fertilize your Pachysandra in the spring with a balanced fertilizer or one with low phosphorous.
12-12-12 or 8-5-5.
Lift the plants to apply the fertilizer to the soil and not the plants themselves.

https://www.gardeningknowhow.com/ornamental/groundcover/pachysandra/pachysandra-ground-cover.htm

Amending the clay soil is an option and good idea to optimize the plants you can grow and the success of what you do grow.
Some options for your area may include Creeping Jenny, Snow on the Mountain, Canada Violet, Beacon Silver.
Pachysandra plantings would thrive in a more loamy, well draining soil.

Here are some links to help you.

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm
https://www.gardeningknowhow.com/ornamental/groundcover/pachysandra/pachysandra-ground-cover.htm
https://www.gardeningknowhow.com/special/shade/ground-covers-for-shade.htm
https://www.gardeningknowhow.com/ornamental/groundcover/creeping-jenny/creeping-jenny-ground-cover.htm
https://www.gardeningknowhow.com/ornamental/groundcover/snow-on-the-mountain/snow-on-the-mountain-plant.htm