temp_weather
is being overwritten every subsequent time the user inputs information. The inputs before the last one are being recorded and then discarded.
Try something like:
#Ask what the rainfall was for each day
while(x<=weather):
total += input("What is the rainfall for day " +str(x)+"? ")
x += 1
print("The total rainfall for the period was "+ str(total))
Note that I changed "weather" to "total" on the last line because I assumed that's what you intended!