What you are describing is not contracting, but temporary permanent employment, and the tax man would probably describe it as such and expect you to pay PAYE and NI through the employer. Rates of pay in this model are poor – in line with permanent employment but with no benefits and no security. Yes, if you are an employee you will get a P45 each time (and an emergency tax code every time it is late – yikes!)
I would advise against choosing this mode of employment. Why not set up a limited company? It costs £100 to set up, is substantially more tax efficient and easy to close down again. An accountant will charge you about £70 a month (£100 if the do you VAT return). You’ll need to work for no more than two hours to earn his fee.
If you can fill in a timesheet and a excel spreadsheet of what you're spent and earned each month, then you have what it takes to run a limited company. Let the accountant do the rest.
Irrespective of the mode of employment CBR checks are now the norm. Company policies differ (and application of them is varied). Most firms say you cannot start until the check is complete, but many bend their own rules if they desperately need you or if their CRB supplier is taking too long to complete thechecks.