Mollysawally, I don't think the flexible working regulations provide a sample application letter, I could be wrong.
You can use the structure suggested by my company's policy. It is very detailed but will help you think through all the issues from the point of view of yourself and your company and pre-empt questions from your boss:
Confirm that you are making an application under the Flexible Working Regulations.
Have you made a request to work flexibly before? If 'yes', when (must not be within a year) and what was the outcome?
Confirm you are requesting flex working to care for a child, the age of the child (must be below 6) and your relationship to the child.
Any days within 28 days where it is not convenient for you to have a meeting.
Describe your current working arrangements, new proposed arrangements and start date.
Describe the key elements of your role and how you can continue to maintain a high level of service with the new arrangement.
How will you maintain good commmunications when you are out of the office.
How will you meet the criteria for performing well and what evidence can you show.
How will you keep up-to-date with technical developments and training.
What IT/telcomms/remote working technology would you need?
What would you need to change in your working practices to implement the new arrangements.
What flexibility do you have if a client or other need were to conflict with your new arrangements?
What adverse effects would your new arrangement have on external clients, internal clients, your colleagues and the people for whom you work.
What benefits will this arrangement bring for the company and its clients (this always makes me laugh)
How will your boss continue to monitor your work and performance?
What are early warning signs that the flexible arrangements are not working.