When planning a playthrough in one of the many games in the Pokemon series, players often need to take into account what Hidden Machines (HMs) their Pokemon can learn—moves that, while not able to be deleted under normal circumstances, allow players to pass a variety of obstacles. One of the most recognizable HMs players obtain is Cut, oftentimes required to progress early on in their adventures.
If a Pokemon knows the move Cut and players have the corresponding Gym Badge that allows it to be used in the overworld, they can then use the move to chop down certain trees and bushes that are strategically placed to block player progression. In certain games, players can even use Cut to clear patches of grass where players would normally encounter wild Pokemon. Other than this, however, the move doesn't have any special properties, and functionally acts as a slightly stronger version of Tackle.
Yet nowadays, these pesky small trees and bushes are no longer obstacles that players will need to bring certain Pokemon to overcome. Since Pokemon Sun and Moon, HMs were removed completely from the series and replaced with alternate ways to traverse terrain, whether it be through simply expanding from linear routes to giving players access to special Pokemon for certain situations.

What Pokemon Can Learn Cut in Previous Generations?
In Pokemon Red, Blue, Yellow, Gold, Silver, Crystal, Ruby, Sapphire, Emerald, FireRed, LeafGreen, Diamond, Pearl, Platinum, HeartGold, SoulSilver, Black, White, Black 2, White 2, X, Y, Omega Ruby, and Alpha Sapphire, Cut can be taught to a number of Pokemon using HM01. This is normally one of the first HMs that players obtain, as early-game progression is usually limited by small trees that need to be chopped down using Cut.
These are the Pokemon by generation that can learn Cut via HM01, with the new additions added for each generation.
Generation 1 (Red, Blue, Yellow)
Bulbasaur, Ivysaur, Venusaur, Charmander, Charmeleon, Charizard, Beedrill, Sandshrew, Sandslash, Oddish, Gloom, Vileplume, Paras, Parasect, Diglett (in Yellow), Dugtrio (in Yellow), Bellsprout, Weepinbell, Victreebel, Tentacool, Tentacruel, Farfetch'd, Krabby, Kingler, Lickitung, Tangela, Scyther, Pinsir, Kabutops, Mew
Generation 2 (Gold, Silver, Crystal)
Bulbasaur, Ivysaur, Venusaur, Charmander, Charmeleon, Charizard, Beedrill, Sandshrew, Sandslash, Oddish, Gloom, Vileplume, Paras, Parasect, Diglett, Dugtrio, Bellsprout, Weepinbell, Victreebel, Tentacool, Tentacruel, Farfetch'd, Krabby, Kingler, Lickitung, Tangela, Scyther, Pinsir, Kabutops, Mew, Chikorita, Bayleef, Meganium, Cyndaquil, Quilava, Typhlosion, Totodile, Croconaw, Feraligatr, Sentret, Furret, Bellossom, Aipom, Sunkern, Sunflora, Espeon, Umbreon, Gligar, Steelix, Scizor, Heracross, Sneasel, Teddiursa, Ursaring, Skarmory, Raikou, Entei, Suicune, Tyranitar
Generation 3 (Ruby, Sapphire, Emerald, FireRed, LeafGreen)
Rattata, Raticate, Meowth, Persian, Rhydon, Kangaskhan, Dragonite, Celebi, Treecko, Grovyle, Sceptile, Torchic, Combusken, Blaziken, Zigzagoon, Linoone, Nuzleaf, Shiftry, Breloom, Slakoth, Vigaroth, Slaking, Nincada, Ninjask, Shedinja, Sableye, Aron, Lairon, Aggron, Roselia, Cacnea, Cacturne, Corphish, Crawdaunt, Anorith, Armaldo, Kecleon, Tropius, Absol, Bagon, Shelgon, Salamence, Metang, Metagross, Latias, Latios, Groudon, Deoxys
Generation 4 (Diamond, Pearl, Platinum, HeartGold, SoulSilver)
Turtwig, Grotle, Torterra, Chimchar, Monferno, Infernape, Piplup, Prinplup, Empoleon, Bidoof, Bibarel, Kricketune, Budew, Roserade, Rampardos, Vespiquen, Pachirisu, Ambipom, Drifloon, Drifblim, Buneary, Lopunny, Glameow, Purugly, Stunky, Skuntank, Gible, Gabite, Garchomp, Skorupi, Drapion, Toxicroak, Carnivine, Weavile, Lickilicky, Rhyperior, Tangrowth, Gliscor, Gallade, Dialga, Palkia, Giratina, Darkrai, Arceus
Generation 5 (Black, White, Black 2, White 2)
Snivy, Servine, Serperior, Oshawott, Dewott, Samurott, Patrat, Watchog, Purrloin, Liepard, Pansage, Simisage, Pansear, Simisear, Panpour, Simipour, Drilbur, Excadrill, Sewaddle, Swadloon, Leavanny, Scolipede, Petilil, Lilligant, Basculin, Sandile, Krokorok, Krookodile, Dewbble, Crustle, Archen, Archeops, Zorua, Zoroark, Sawsbuck, Emolga, Karrablast, Escavalier, Joltik, Galvantula, Ferrothorn, Eelektross, Axew, Fraxure, Haxorus, Cubchoo, Beartic, Druddigon, Pawniard, Bisharp, Bouffalant, Rufflet, Braviary, Vullaby, Mandibuzz, Heatmor, Durant, Cobalion, Terrakion, Virizion, Reshiram, Zekrom, Kyurem, Keldeo
Generation 6 (X, Y, Omega Ruby, Alpha Sapphire)
Chespin, Quilladin, Chesnaught, Fennekin, Braixen, Delphox, Froakie, Frogadier, Greninja, Bunnelby, Diggersby, Pancham, Pangoro, Espurr, Meowstic, Honedge, Doublade, Aegislash, Malamar, Binacle, Barbaracle, Clauncher, Clawitzer, Helioptile, Heliolisk, Sylveon, Hawlucha, Klefki, Phantump, Trevenant, Noibat, Noivern, Xerneas, Yveltal, Volcanion
Starting with Pokemon Sun and Moon, HMs were removed entirely, and have not returned in the main series since. Whether they were replaced with secret techniques or alternate maneuvers to get rid of small trees, or these obstacles removed completely, it has rendered the move Cut almost entirely obsolete. From Pokemon Sun and Moon to Pokemon Sword and Shield, only Farfetch'd (by extension, Sirfetch'd) and Kartana can learn Cut, both via level up.

What Pokemon Can Learn Cut in Pokemon Scarlet and Violet?
Following the removal of HMs from the series, Cut is no longer a move that can be learned by any Pokemon in Pokemon Scarlet and Violet. Since Farfetch'd and Kartana can not be obtained in these titles, Cut has not been programmed into them.