I think the key things are to be clear what you are requesting, avoid unnecessary details and focus on explaining why/how your request will not negatively impact the business.
So I would start by simply saying something like "Dear [manager], I am writing to make a flexible working request. I would like to reduce my working hours to 30 p/week with effect from ...
Then I would briefly say why you want to reduce hours. Just say something simple like 'this will assist me to manage childcare responsibilities' - sometimes people go on massive long sob stories about why they need the flexible working, and IMO this is unhelpful as the manager isn't supposed to be assessing who is 'deserving' of flexible working, in theory everyone is equally entitled to consideration, whether they want the time to look after the 20 disabled orphans they have adopted or to lounge on the sofa watching Jeremy Kyle! But just in case your manager feels its important I would just touch on your circumstances.
Then the main part should be you explaining why you feel this won't have a negative impact on the business as this is the main evaluation criteria, so cover things like whether you believe all your work can be covered in 30 hours or whether they will need to cover it elsewhere, if the latter how the work can be covered etc (e.g. if you know other part time staff would be keen to increase their hours to cover the day you won't be in that is worth mentioning). If you are able to be somewhat flexible, e.g. you would be prepared to have 2 mornings/afternoons off rather than one full day, or could be flexible with what day you have off that would be good. I would also always say (if at all possible) that given sufficient notice you would be prepared to swap your day off or come in on your day off if there is a crucial meeting or training etc. that can't be rearranged. Also always say you would be happy to agree to reducing your hours on a trial period basis at first so any issues or concerns can be tested first.
Sign off by saying you would appreciate the chance to discuss the request with them in person and thanks for their consideration etc...
Good luck!