Running the site I'm currently working on through The W3C Markup Validation Service it brings up 34 errors! Mostly does not allow element "p" here. I see the validator has the option to Cleanup Markup with HTML tidy.
What's the recommendation here for sorting problems out please.
It depends. What errors does it give you?
HTML Tidy wouldn't be much use for a Joomla site as there'll be a lot of server side processing. I'd only ever use it to tidy up a static html page, and even then there's always the chance that it'll send your website all over the place.
I see there is a Clean-up Messy Code icon in the admin page editor. Is that safe to use? Or am I best doing my editing line by line in notepad. Sample entry found by W3C:
Line 249, Column 50: document type does not allow element "p" here; missing one of "object", "applet", "map", "iframe", "button", "ins", "del" start-tag
…Normal" style="margin: 0cm 0cm 0pt;"><span style="font-family: "Verdana&✉
The mentioned element is not allowed to appear in the context in which you've placed it; the other mentioned elements are the only ones that are both allowed there and can contain the element mentioned. This might mean that you need a containing element, or possibly that you've forgotten to close a previous element.
One possible cause for this message is that you have attempted to put a block-level element (such as "<p>" or "<table>") inside an inline element (such as "<a>", "<span>", or "<font>").
I get that error for my site too. For me, it's from the html I use to centre the highslide slideshow on each of my news items, but I'm not sure of a way around it (or at least haven't investigated it properly yet).
I've never seen the Clean-up Messy Code icon, although I use JCE rather than the default TinyMCE. It might be worth a try, as long as you back up what you have there currently beforehand (easiest method is just to copy and paste it into Notepad, and then copy and paste it back into the editor when everything goes wrong).