Firstly - "I am not qualified to offer tax advice" !!
If you have been paid by two employers during this tax year ( from 5 April 07) then you will have to declare two employments. Whether you pay tax is a different question, based on how much you were paid in this tax year and whether you have any additional allowances above the standard personal allowance (
£5225 in 2007/8). If you are doing only part time work, I would expect they will not be taxing you at source, but having effectively had two concurrent jobs you will probably need to ask for a self-assessment in April 08.
BTW, do the library know you are planning to submit a May timesheet in August - was this agreed with their payroll people?
HTH