Q. If power is increased during a turn while maintaining the stalling angle of attack
A. The radius of turn will remain the same and the rate of turn will increase
It gets a bit more complicated than that simplistic idea. FIrst, perhaps have a read through the following post on PPRuNe, noting that some of the posters are extremely expert FJ pilots, test pilots and demonstration pilots.
www.pprune.org/military-aviation/638681-...ml?highlight=viffing
If you like to search in PPRuNe, there are more than a few threads which talk to the points which your thread revolves around.
I guess, basically, you need to keep in mind a few things -
(a) the highest performance is at the cornering speed which is where the stall line intersects the limit load factor. This is what most pilots (incorrectly) consider to be the design manoeuvring speed - it may be, but it also may not be, depending on the the aircraft and what the OEM designer chose to adopt as the manoeuvring speed for the design. That's another story, though.
(b) the big problem with manoeuvring at this point is that the drag imposes an extremely high thrust requirement. FJ ACM employs reheat (afterburning) to get all the thrust the pilot can find but, as you will see from the cited thread, that still may not let the pilot operate for more than a short period in that regime.
(c) often, the best the pilot can achieve is some useful pointing of the lift vector with associated ACM benefits.
(d) for light aircraft, I can't bring any aircraft to mind which can manoeuvre near that point without the drag penalties slowing the bird down drastically and the turn performance then goes out the window.
(e) if you are operating at a more sensible point for the aircraft, you may get some instantaneous benefit with increasing power (the Harrier used viffing to useful effect in combat in this manner).
(f) quite some years ago, I was following a rather erudite thread in PPRuNe and asked a question along the lines of how much value was reheat in high performance manoeuvring. I haven't been able to locate it at the moment but it probably was in one of the Shoreham Hunter mishap threads. As best I can recall, the response (from a very appropriate poster) was along the lines that reheat could provide a decrease in loop radius of around 1000 feet. That probably goes some of the way to answering your question ?
the question states an increase in *power*
The question expected that you would extend the term "power" to mean "thrust" which is what is important.
if this was the case why would the rate of turn then increase also?
(Aside - if you can maintain radius but increase speed, or reduce radius and maintain speed, you are going to go around the circle quicker, hence a higher turn rate)
The only reason you would be playing at corner speed would be to achieve maximum performance (presuming you could get enough thrust to fix the drag problem). A principal driver with maximising turn performance (typically for ACM considerations) is to increase turn rate and/or decrease turn radius and, as you can read from the link, with enough grunt, you can achieve that even if only for a short period.
The other consideration is that, for this area of the envelope, you are operating at higher alpha and, if you have some spare thrust, a component of the thrust vector can be used to unload the wing which may allow you to pull a bit harder to get back to the limiting aerodynamic point. One of the downside points is that this will still increase the g-loads on the structure due to the centripetal acceleration variation considering speed and radius so there is still no free lunch.
Now, if you are operating at a bank angle well below that for the limit load factor (to accommodate a light aircraft's rather abysmal engine output capabilities), if you are able to increase power (and thrust), then you should be able to pull a fraction tighter which will probably give you a small reduction in turn radius.
So, for the question, it depends on the aircraft's capabilities and particular design characteristics. Probably not anything more than a discussion point for light aircraft but, as you can see from the linked thread and others of similar topics, it is extremely pertinent for the FJ ACM crowd, albeit that, for the modern jet designs, ACM probably is going to go the way of the dodo.