Weak pinkie? Uh-oh. I am wondering about your wrists. It sounds like you may be dropping them if your pinkie is weak. It is really important for your wrists to be in line with the keyboard and level with your elbow... or slightly lower than your elbow... while your hands & fingers then drop below your wrist. Of course, this has to feel like a comfortable, relaxed position with good back & leg posture. (Feet always square & flat on the floor.) Nothing should feel forced. Your position will come from the entire lower arm, but the wrist shouldn't "break" and the fingers should be completely mobile and wiggly.
Lifting your wrists (another way of putting this is not allowing them to bend) puts your hand in the right position so that your pinkie and all your fingers can be strong and extend fully. It gives your fingers the freedom to be strong since they "hang" from your elbows via your wrists. It is also important to curl your fingers. At first you may want to exaggerate the curl (which would be the only time you'd feel less than relaxed). Just don't play with flat fingers. Tell those fingers you mean business! The position is not the same hand position as you would use on a computer keyboard where your wrists are dropped & probably resting on a pad. It may be that you'll need to be sure your piano keyboard is at the correct height, or else raise your seat so that you can approach your piano with the correct position. Piano stools are made to be raised or lowered for this reason.
To strengthen all your fingers, & especially your pinkie, I hope that you will consider doing scales, Craven. Nothing is better for strengthening your fingers and developing a good strong touch. Scales require that every finger to do their job and that's part of why they're important.
Start with the left hand, first octave of middle C. Thumb on mid.C, pointer on D, middle finger on E, then, sneak your thumb underneath the middle finger & play F with your thumb. Continue GABC with your 2,3,4,5 fingers... and return exactly the same way, but this time lifting the third finger across the thumb, (and continuing 3,2,1). Go slowly, make sure you hit every note clearly & try to make each note sound exactly the same. (Boring, yes.)
You can also practice a scale without your piano keyboard. Any old table will do. The point is to get your hands and fingers totally set in the correct pattern of movement, 'cause it will be used over & over.
After you've done about ten octaves, up and back, and assuming you feel confident with them... here are some variations so you don't get bored. Play quietly up & loudly back (Can your keyboard allow different levels of sound?) Play every other note loud or soft. Play your octaves with different rhythms. Swing it with short, long, short, long, short, long as you go up, and then long, short, long, short, long, short, coming home.
There are scales for every single note, including all the sharps and flats, but just practice with C for now. Playing a song in any key is always easier after you've played a few scales in that key. It is much easier to hear and play the right notes. (That's the other really good reason to do scales.)
For the right hand, you'll start with the pinkie on the C below middle C, and continue 4,3,2,1, then lift your middle finger up and over the thumb & continue, 3,2,1. Return down the scale the same way, 1,2,3, 1,2,3,4,5. Your thumb goes under your third finger.
When you get comfy with each hand separately playing the scale, then you'll want to play them together. It's really good practice to get yourself thinking with two hands with scales (which are sort of mindless) because eventually you will be playing songs with two hands.
(Maybe I'm telling you stuff you already know and your piano lessons have already got your doing scales? I hope so. Otherwise, please start. Five or ten minutes of scales during your practice will be so good for you. You'll thank me in five years.)
Here's an interesting website that describes the body stress of piano playing: http://www.musicandhealth.co.uk/stress.html