Yes you can transfer your existing PAYG phone number to a new phone, either contract or PAYG. In order to do this you will need to request a Port Authorisation Code (PAC) from your current network provider.
What you do with the PAC code will depend on the particular phone you are looking at purchasing. For contract phones many retailers and networks may require you to provide your PAC code during the purchase process in order to transfer your number over and will not allow you to do so after the order is completed. Because of this you should check the information provided on the website of your chosen retailer to see what their rules on number porting are.
Other retailers may not require you to provide the PAC code during purchase but may have a limited time period (7 days for example) after the order is completed for you to request to have a number transferred. Transferring a phone number to a PAYG phone will usually not have any time restriction and you will not be required to do so during the purchase process.
When transferring a phone number from a PAYG phone to another phone you will lose any credit you have on your PAYG handset as this cannot be transferred. Because of this you should make sure that you use any outstanding credit on your phone before your number is transferred to a new one.