I think your xpath is not quite right. Here's what works for me:
from splinter import Browser
import time
URL = 'https://mail.google.com/mail/u/0/?shva=1#trash'
with Browser() as browser:
browser.visit(URL)
username = browser.find_by_id('Email')
username.fill(...)
password = browser.find_by_id('Passwd')
password.fill(...)
button = browser.find_by_id('signIn')
button.click()
time.sleep(5)
browser.visit(URL)
empty_button = browser.find_by_xpath("//div[5]/div[2]/div/div[2]/div[1]/div[2]/div/div/div/div[2]/div[1]/div[1]/div/div[2]/div[3]/div/span")
empty_button.click()
Though, you should think about simplifying the xpath expression, xpaths with absolute paths are too fragile.
Hope that helps.