Is there an app that books the job in when the deposit is paid?
Yes. InstaQuote does it. Your customer accepts the quote and pays the deposit by card on their phone. The job then drops into your next free days on its own. If the date doesn't suit, the customer moves it themselves from the same link. Booked jobs are copied into your Google Calendar if you connect it.
Plenty of apps stop at the deposit. The money lands, then you still have to ring the customer, find a day and type it into the diary. That's the bit that eats your evenings.
With InstaQuote the deposit is the booking. It looks at the days you already have booked in InstaQuote, picks the next free ones and tells you and the customer. Booked jobs are copied into your Google Calendar if you connect it. It books round the jobs in InstaQuote, so put days off in the InstaQuote diary.
For a homeowner it keeps to their 14-day cancellation period, unless they've asked you to start sooner. The customer can still pick a different free day from their link, and you get told when they do.
No other trade app we checked on its own website does all of that from a quote you wrote. You can see what each one says on our compare pages. InstaQuote is free for 7 days with no card, then plans start at £49 a month.