Whitney's Clefairy isn't anything too special. It has a few tricks up its sleeve, but it's
relatively weak all things considered. Mimic lets Clefairy use the move you last used
against it any time it wants in the fight, though you should hopefully be knocking it out
in 2 hits anyway, so it shouldn't really matter. It can also use Metronome to use any move
at random, which can make or break the fight in a few ways.
That's not the Pokemon you have to be concerned about, though; her real threat is her
Miltank. It's at a sturdy level 19 and totally ready to go. It has Stomp and Rollout in
its attacking arsenal, with Attract and Milk Drink supporting it. Stomp is very powerful
coming from the strong Normal-type cow, plus it has a chance of causing you to flinch. Not
good! Rollout is also especially painful if it hits more than 2 times in a row. One of the
best things you can do against her Miltank is to lower its accuracy with Smokescreen or
even Flash. That will hopefully cause Rollout  which doesn't have awesome accuracy in the
first place  to miss before it gets out of control, resetting its power.
Male Pokemon are at a huge disadvantage if Miltank decides to use Attract, turning each of
their attacks into a virtual coin flip. The best thing you can do to get out of it is to
just switch out for another Pokemon so the effect goes away. Risking that coin flip every
turn is often too much risk than what you want to shoot for, so switching is a safer
option.
The worst part is that Miltank  like Falkner's Pidgeotto  can restore 50% of its max HP
with Milk Drink. It has a good amount of HP and Defense, too, meaning it is a real pain to
take out. If you try affect it with a status condition, it will heal it with a Lum Berry
(though you may be able to try again next turn after the Lum Berry has been consumed).
Plus, Whitney also has a Super Potion she's just dying to use! Talk about a tough fight!
Items like X Attack and X Defend, which can be purchased in the Dept. Store, can really
help turn the tide in your fight against her. Use a few of those, throw in a Flash or
Smokescreen for good measure (otherwise Rollout will crush you if you are solely trying to
set up), and then take her out as quickly as possible. Trading a Drowzee for a Machop is
also a smart move, because Machop can hit pretty hard against Miltank, particularly if it
gets a buff or two from the X Attack and X Defend. Just watch out for Attract, as most
Machop are male!
After beating her, she will cry and won't give you the badge. What a sore loser!
Thankfully, as you try leave, Lass Asuka will tell you that she's a crybaby and can't
handle losing, so you should talk to her again and try rub it in her face more. Nah, I'm
kidding. But you do need to talk to her again to get your badge.
Whitney will now give you the Plain Badge, along with TM45 (Attract). Oh that Attract,
that awful Attract. It makes for a nice move to teach some of your Pokemon, but it's
useful only against Pokemon of the opposite gender. There seem to be a lot more male
Trainers (which means male Pokemon) in the game, so it's better taught to a female Pokemon
if you want the most out of it.