Through applying for a passport this month and seeing all the updates from people I would say that 90% of the time ‘processing’ means there’s an issue with the passport and it’s likely put in a pile for manual processing. The passport office don’t know if or when you’re going on holiday, so these passports aren’t triaged or prioritised. Getting in touch with them seems the best way to get them picked out and worked on, otherwise you’re just depending on someone to work their way through the pile and to get to yours. So Definitely get in touch with them if you’re on holiday soon, and if you applied 6+ or 10+ weeks ago you have a good case to push for priority.