I have what looks like perennial growing in a shady(ish) section of my lawn. I'd like to stretch this along the whole shady section to replace the weeds that are growing there. Could I simply pull out a clump, cut it up into 4-5 smaller sections and plant?
To figure out whether transplanting will spread this particular grass, of which I don’t know the specific identity, you’ll have to find out how it reproduces. You can discover that by experimenting and/or examination.
Grasses variously reproduce from seed, stolons, or rhizomes. Many grass species can reproduce via more than one of these plant parts. In temperate climates, grass rhizomes and especially stolons are most common and easiest to observe in spring, but it’s quite possible to spot them any time. Googled images of ‘grass rhizome’ / stolon may help as you check your grass.
Stolons exist above the ground surface. They extend from the parent plant and are capable of taking root. Once rooted and established they do fine even when severed from the parent plant. Indoor ornamental spiderplant “babies” are stolons looking for soil. Another way to check whether it makes stolons is to observe whether the grass makes thatch, which when thick and tangled often is mostly dead stolons. From the photo this grass appears not to make stolons, but check.
Rhizomes have similar functions to stolons but spread underground instead. So dig up a shovel-size scoop a couple inches deep. Shake thoroughly or even rinse the roots. Look for structures that resemble stubby-tipped roots extending mostly horizontally away from the parent plant. They may be segmented and they rarely branch. You’ve probably seen rhizomes of grasses locally called “quack” or “crab.” Particularly when young, rhizomes usually are lighter color than regular roots. They may extend many inches from the parent.
If your grass reproduces in part from stolons or rhizomes, it can be extended by transplanting and tending plugs. Otherwise it can’t.