Determining if my artificial neural network needs additional layers
There isn't any hard and fast rule to determine that how much hidden layers would work. In your case, extra layer won't help that much, but extra features can (I'm pretty much sure). Weather forecasting, as far as I know, isn't something which can be determined using only 4 features (considering your input neurons). You can always come up with more and more features, which effectively help in predicting the weather. So, I'd suggest to look for more features (about 10 to 15 useful features), and come up with the following configuration of the network:
1.) Input: Number of neurons in he input layer should be equal to the number of features you calculate for each data entry.
2.) Output: If you can categorize the weather in, say, 5 different categories (sunny, rainy, foggy, stormy, meteor shower), then there can be 5 output neurons.
3.) Hidden: Number of neurons in the hidden layer is normally the mean value of the number of input, and output neurons. You can also have same number of neurons in the input, and hidden layer.