Loose Change:
OK, I gotta question, just exactly what kind of training are you after that maybe wasn't adequately covered in your L's course in enough depth?
Thanks for that, Loose Change.
Well anything really! I'm not saying the basics weren't covered. I felt the L training was really pretty good for someone who started with very little idea. The instructor was great and I did learn a lot. However, it's silly for me to think that 7 hours of very basic training, on a small area of flat bitumen, in a very controlled environment, in good weather, will equip a person (me), for riding on all sorts of roads, terrains, traffic and weather. Because I don't have biking friends around me who can focus their time on helping me whenever I need it, I'm happy to pay for someone to give me further help. It makes sense. (Stayin' alive, stayin' alive!). 
As an example, the problem I have with opening the throttle as I'm trying to brake is one very important issue that I need to resolve - and quickly. Forums like this are invaluable, and I'm doing my best to implement the advice so kindly given here, but having an experienced trainer actually showing me is a different thing entirely.
It's hard to know 'exactly what kind of training' I'm after, as I'm new to all this and really don't know what I need to know, until I'm in a situation where I need to know it!! I know how I learn best and it isn't by reading, but by asking, seeing, and doing, so that's why I think further training is the go for me anyway.
I guess, so far, the things which concern me are the braking issue, going around bends and staying upright, wet weather riding, and I suppose a host of things which will undoubtedly come with more experience, but would certainly be helped with a professional trainer.
I think, for experienced riders, it might be difficult to reflect on the time when you were like me, and not confident on the scooter. It's a bit like me, with 35 years driving experience. I really don't think too much about it, I just drive. Pretty much everything I do is so completely natural to me, I only have to watch the road and other users and anticipate what might happen. But being on a new type of vehicle, the scooter, I'm not only trying to do those things, but also trying to get to know and master the machine, be aware of the centre of gravity of the scooter, the environment (which is now a much more critical factor), the weather and all sorts of things I don't actually think too much about when in my car. It really is such a different experience. Being older, too, adds it's own element to the mix, as we simply don't learn as well at this age, and are much more aware of the possibilities of bad outcomes. I learned to drive when I was 17 and hadn't lived much. At 52, and being a nurse, I have seen many things most of us never want to see. I've lost many friends in accidents - some of them just horrific. I'm aware this can happen to me. So, I am just trying to do what I can to minimise the chances of me hurting myself or anyone else.
I wish I could be more specific, Loose Change, but as I say, I'll only know what i need when I need it - and that may be too late.
But all up I just want to have fun and stay safe! 
Don't take life too seriously. You'll never get out of it alive.
2011 - Pearly White Kymco Like 125cc
2012 - Giallo Lime (yellow) Vespa GTS 250ie
2016 - Shine Red, Aprilia Scarabeo 200ie