Basically the reverse Cousin Oliver (https://tvtropes.org/pmwiki/pmwiki.php/Main/CousinOliver)/Scrappy-Doo (https://tvtropes.org/pmwiki/pmwiki.php/Main/TheScrappy), where the addition (or in some cases, replacement) of a character gave the show some needed energy.

The Big Bang Theory - Both Bernadette & Amy Farrah Fowler's additions to the main cast made it better by not having Penny be a typical foil to the four guys' nerdiness.

Steve Urkel, Waldo Faldo, Myra Mounkhouse - "Family Matters"

Pam Tucker, Charmaine - "The Cosby Show"

Georgette Franklin and Sue Ann Nivens, the Mary Tyler Moore Show.

Pebbles and Bamm Bamm, the Flintstones.

This is a tough category, since so many additions flop.

Max Horvath - One day at a time

Barnabas Collins, Dr. Hoffman, Sarah, Angelique - Dark Shadows
Uncle Arthur, Serena, Maurice, Aunt Clara, Dr. Bombay, Phyllis, Tabitha - Bewitched
Sue Ann Nivens - Mary Tyler Moore
Batgirl - Batman
Gomer Pyle, Goober Pyle - Andy Griffith
Tommy Chong - That 70s Show
Lilly Tomlin - Laugh In
Sister, Gin-Gin, Amanda Bellows - Jeannie
Annette (Gina) - Make Room for Daddy
Mr. Furley - Three's Company
Chachi - Happy Days (debatable)
IRAC, ROVER - Wonder Woman (debatable)

Reverend Jim (Taxi).

Reverend Jim (Taxi).

Ah, you beat me to it.

I guess Seven Of Nine on ST: Voyager.

Cliff Clavin, Frasier & Rebecca on Cheers.

Monroe on Too Close For Comfort (he added something to the show)

Larry, Daryl & Daryl and Peter Scolari's character on Newhart.

Most of the other characters on Last Man On Earth.

Star Trek: Voyager really improved in season 4 when Seven of Nine replaced Kes.
Antonio was a good addition on Wings.
Arthur Dietrich was a good addition on Barney Miller.
Benson really improved when Clayton Endicott III replaced Taylor in season 2.
Blue Bloods- Adding Marisa Ramirez and Vanessa Ray to the cast in season 4 as Danny and Jamie's partners (and now Jamie's wife in Vanessa's case)
Adding Bill Gannon to the color seasons of Dragnet improved it from the 50s version.
Early Edition was always a great show, but adding Erica and Henry Paget made season 3 my favorite. But then they were gone too soon in season 4.
Georgie was a definite improvement over Mallory on Heartland.
Law and Order started out as a dull show until Jerry Orbach became the star of the police section. And it had several other good cast members that came like Sam Waterston, Jill Hennessy, Benjamin Bratt, Jesse L. Martin, S. Epatha Merkerson, Fred Dalton Thompson, Elisabeth Rohm, and Angie Harmon.
At the very beginning of Mad About You, Paul's best friend was a dull character named Selby. He was replaced halfway through season 1 with cousin Ira, who stayed for the rest of the show. Also adding Cynthia Harris to recur as Paul's mother.
M*A*S*H- It was a great show every season. But Harry Morgan, Mike Farrell, and David Ogden Stiers were excellent newcomers who freshened things up. M*A*S*H handled cast changes really well.
Monk- when Natalie replaced Sharona
Murphy Brown (original run)- adding Stan and Andrew Lansing as recurring characters. And Grant Shaud was great at his role, but so was replacement Lily Tomlin.
Newhart- adding Larry and the Daryls
7th Heaven- Kevin Kinkirk was a lot better than any of Lucy's previous boyfriends.
Family Matters- Waldo Faldo was a lot funnier than Eddie's first best friend.
The Waltons- While the later seasons had a lot of flaws like Rose's family, fake John-Boy, and Curt actually being alive but a terrible person now, the addition of Cindy as Ben's wife was a really good one.
Young Sheldon- when excellent actor Wallace Shawn was added to the cast

Re Larry and the Daryls on Newhart--they were on there starting with episode 2. But they probably weren't on recurring status until later.

More good additions to Andy Griffith -- the fun girls, Ernest T. Bass, the Darlings. (Not so good--Helen Crump.)

Diff'rent Strokes--I would call Sam an OK addition (except when he sang), and Dixie Carter's Maggie a good addition. Mary Ann Mobley just wasn't a good choice for the role in the final season. The problem wasn't the added actors/parts, it's that most of the scripts really headed south, particularly in the 8th season.

Re Larry and the Daryls on Newhart--they were on there starting with episode 2. But they probably weren't on recurring status until later.They became more prominent when Kirk left and they ran his cafe, only occasionally appeared before season 3.

Speaking of Newhart, Peter Scolari and Julia Duffy were good additions. Jennifer Holmes' Leslie was too normal for the cast of loonies who surrounded Dick and Joanna.

Oh, here's another one - Conchatta (sp?) Ferrell (Berta) on Two and a Half Men. She only appeared as recurring, a few times in the first season. But picking her up was an excellent add.

The Facts Of Life-Jo and Beverly Ann Stickle
The Cosby Show-Sondra and Elvin
Star Trek:The Original Series-Chekov
Mister Ed-Gordon and Winnie Kirkwood
Adam-12-Officers Ed Wells and Jerry Woods
T.J. Hooker-Officers Stacy Sheridan and Jim Corrigan
Hunter-Sporty James and Captain Charlie DeVane
Little House On The Prairie-The Garvey Family, Adam Kendall, Almanzo Wilder, Nancy Oleson and Jenny Wilder