Hello Bud , I have worked for a few agency’s in the past bear in mind wasn’t trucking work ( which shouldn’t make any difference as I was doing General Warehousing ) and every time I’ve started up with one they’ve never asked me to give them a P45 or P60 
If they want proof of your UK residency then the passport should be all they need.
I’ll be honest every agency, I’ve encountered are right b’tards, They are not interested in anyone’s concerns or problems about the work they just want the work they’ve assigned to be done.
Here’s a few tips to cover yourself from my own personal XP:
Whatever hours you do make a note of it, record it down, Agency’s like to under cut people sometimes , not always , but they sometimes do. If it so happens then harass them as they wont be in a hurry to pay you if they do mess up. The company you work for should have a record of what you have worked which you can use.
Before you leave for work whatever time of day it will be , 30-60 mins before you leave for work , Ring the agency to check if you are going in! ( This one is a biggy imo).
So many people have gone to the place of work to find out they are not needed and the agency didn’t ring to let them know, so people have wasted their time and fuel going there. At the end if your first day, Give the company your phone number and ask them to call you instead of the agency so you know straight from the horses mouth if you are needed the following day.
I got told these two from one of my friends who does trucking and did agency in the past:
If an agency sends you to any place , Go over the Rigid or Artic like a forensics officer looking for DNA, Check it carefully as it might not be maintained to the correct standard, The last thing you want to do is to get nicked by VOSA for driving a bad truck and risk the consequences. If in doubt , pull out and find other places 
No matter what the agency says, you make sure you stay legal, you need your breaks for your own safety and other road users, don’t be pushed to a point you could loose your licence or end up killing yourself. Agency’s only care that the work is done after all , they dont know the rules truckers have to abide by.
Not trying to scare you/put you off or sound morbid but I am assuming you are new to Agency work and rather fill you in. Just be aware mate 
However these things could not happen and it’ll be smooth sailing , if so congratulations on getting some trucking work and most of all " experience " 
jrt:
Plus I’m looking just for part time work to start with, like weekends or the odd week day, is this going be some thing agency’s don’t want ?
Agency’s operate differently but a lot of the time they need people for last minute jobs where people haven’t been able to go in, some people like it this way and other dont.