Of course, sorry.... my omission. It's a Phillips Hue system and the starter pack included a Bridge, a 4 button remote switch and 2 Phillips Hue White Ambiance lamps. I downloaded the Phillips Alexa app onto an iPhone 6s, which was from where I did the initial setup .....successfully.
OK..there is a Philips Hue app, and there is an Alexa app...but there is no such thing as a "Philips Alexa" app. The Hue app come from Philips and is for configuring/controlling your Hue Bridge and the bulbs that are controlled by it. The Alexa app is from Amazon and is for cofiguring/controlling your Alexa account and Echo device(s). So you need to differentiate between the two so we know what you've done (or need to do) in which app.
That was Thursday 28 Sept. since then I decided to add a third lamp but the system doesn't 'discover' (detect) a third lamp!
I'm assuming that by "lamp" you mean "Hue bulb". If not then please clarify what it is that you actually bought and attempted to add to your home automation ecosystem. When seeking help with technical issue accuracy and non-ambiguity in terminology is very important, so please try to be more clear in your descriptions.
I initially thought it was faulty lamp logic, so bought another but that also is not detected.
Not detected by what? The Hue Bridge, or the Echo/Alexa app? Remember that your bulbs are ultimately being controlled by your Philips Hue Bridge, so that is the configuration you need to get working first. Then, once that is done, you can attempt device discovery via your Echo (or the Alexa app) and, if everything is working correctly, Alexa will learn about your bulbs from the Philips Hue Bridge (albeit indirectly via the Philips cloud service) and provide you with voice control over them.
Also....neither of these additional lamps respond to the wireless switch, which was included in the start-up pack.
This tells me that you're not successfuly going through the procedure required to give your Hue Bridge control over your new bulbs (assuming you bought compatible bulbs, which we don't know yet because you chose to refer to them as "lamps" for some reason). As such Alexa and your Echo are irrelevant at this point. You need to get the bulb <-> bridge configuration working first.
I've also swapped lamps around in different lighting fittings, where 2 lamps work but not three!
I'm sorry, but I have no idea what you're saying here.