On a cloudy day, what will be the output of a GPS device?
There is no definitive way of knowing if cloudy weather affects the GPS's performance. Additionally, if we were to assume it does affect it, there is no way of knowing for sure what effect it has on the output.
GPS performance also changes from device to device. (I speak purely in this context of Android devices). The 1st generation Samsung Galaxy S is know for it's terrible GPS performance. To the extent that a cursory search on XDA will reveal several attempted fixes plus a few hardware mods too.
In my opinion, however, your statement that cloudy weather affects the GPS performance is a common myth among plenty of GPS users. The first source clearly states that the loss when using GPS in cloudy conditions is a mere 2db. Hardly enough to affect it's performance as you state it does in your app.
Extract from Source 2:
The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites.
A possible solution: If you think GPS is affected by the weather conditions, you could consider using the Network provider to ensure seamless performance in your app.