Whether you can claim this as an expense hinges on whether it is a repair or a replacement.
To be a repair, you must not have replaced the entire fence. It is not clear from your message if that is the case, but in my experience you don't get a lot of fence for £800 so I would assume not.
Also to be a repair, you must not have improved the asset. I would argue in this case that although the new fence has been built more soundly than the old one there has been no improvement in its function or utility, only its durability, but HMRC could well argue against that.
Finally, assuming you believe that the expenditure passes those two tests, only part of the expenditure is allowable, because the house is your home as well as an asset used in the business. The normal method of apportionment of household expenses is based on the number of rooms used exclusively for the business as a proportion of the total number of habitable rooms (living and sleeping rooms not including kitchens, bathrooms, utility rooms etc.).
So if you have three bedrooms, a dining room and living room and use the dining room exclusively for childminding, in general you can claim 20% of your household expenses, so in this case £160.
You could try an argument that the garden is not an anciliary part of the home but performs a special function for the business as a play area and so the normal habitable room apportionment is inappropriate. The fence does fulfil the normal function of marking the boundary of your home, so you can't argue the cost is wholly attributable to the business, but it also fulfils a different function of helping to protect the safety and security of the children. On this basis a 50:50 apportionment seems reasonable, although HMRC may take a different view on an inspection.
There is not usually a black-and-white answer where tax is involved I am afraid!