It is up to you to decide when you are going to work, if you decide that level of rain is workable, then work.
We don't work in pouring rain, but heavy drizzle, yes. If customer says anything, normally say this is british climate, when do you expect us to clean them.
Fit new customers into your round, and do them after the house nearest, don't make a special trip for them. or else they will be out of sync. with the rest of your round.
Remember it is a round, so you should be working round the customers one by one, in the order you have arranged, don't be drawn into coming back for odd ones that don't want them done that day.
Don't let your round run you, you should run your round.