We have the technology to quickly and efficiently map the locations of these islands (satellite imagery), and with enough observation we could likely model their drift. The issue is, even current atmospheric forecasting models are not very accurate beyond a week at best, so the models would be guesses. That being said, if we assume that these models are sufficiently accurate, and that the islands maintain at least a somewhat constant size and shape (erosion would have to be factored in), we still run into a few problems.
We would need to carry out studies regarding the effects of insolation refracting at higher altitudes than the Earth's surface and whether or not more or less radiation is trapped by GHGs because of this. What ends up happening in this case is that likely (depending on how high they are) we actually would lose a little more radiation to space than currently, and consequently the rate of global warming slows very slightly. However, the bigger issue for me is the effect of shadows of these islands. A Corsica sized shadow is a pretty big shadow, and that's only going to occur at noon when the sun is directly above Corsica-Sky-Island. Any other time the sun is up the shadow could be much bigger (like how at low sun angles our shadow can be much taller than us). This would have a drastic impact (although not immediately serious) on vegetative productivity and we would probably end up with a considerably less plant life in several years (depending on quantity of islands). This would cause less CO2 to be recycled and in turn speed up the rate of global warming.
We also have to account for how these islands are floating in the first place, if you are talking some theoretical super-light mineral composition that is less dense than air then we have relatively fast moving drift and have to figure out insolation absorption rates and refraction properties for this new composition to be accurate. If they are normal metals and rock suspended by some sort of magnetism, then we could see very drastic effects on climate. Powerful magnetic fields like that in the atmosphere could theoretically reduce the amount of UV radiation the Earth System takes in and have unforeseen effects on climate. Less sunburns, but also could lower our global temperature considerably.
As far as the islands effects on immediate weather systems like thunderstorms and hurricanes, your guess is as good as mine, but I would imagine we see overall somewhat less average precipitation on the surface as moisture in the atmosphere can presumably collect on these islands. I don't know about weather, that would be the big game changer. Long term climate changes are perhaps easier for me to think about.
So, are we able to predict their influence on Earth climate?
The short answer is yes, but probably not accurately until we have ~20-50 years of observations and hardcore science time.