The code is far from pretty, but it works (at least as long as something goes wrong and it crashes). It gets the current weather information from the OpenWeatherMap API, and displays the icon and temperature, alternating them every 4 seconds. The particular prototype shield I used for testing also has 2 buttons, but they are not used by the code.

import utime
from machine import I2C, Pin
from ht16k33_matrix import Matrix8x8
import json
import urequests


API_KEY = "generate your own"
CITY = "Zuirich"
ICONS = {
    '00': (
        0b00000000,
        0b01000010,
        0b00100100,
        0b00011000,
        0b00011000,
        0b00100100,
        0b01000010,
        0b00000000,
    ),
    '01': (
        0b00010000,
        0b01010010,
        0b00111100,
        0b00111111,
        0b11111100,
        0b00111100,
        0b01001010,
        0b00001000,
    ),
    '02': (
        0b00010000,
        0b01010100,
        0b00111000,
        0b11111110,
        0b01000100,
        0b10000010,
        0b10000001,
        0b01111110,
    ),
    '03': (
        0b00000000,
        0b00000000,
        0b00000000,
        0b00111000,
        0b01000100,
        0b10000010,
        0b10000001,
        0b01111110,
    ),
    '04': (
        0b01100000,
        0b10011000,
        0b10000100,
        0b01111000,
        0b01000100,
        0b10000010,
        0b10000001,
        0b01111110,
    ),
    '09': (
        0b00111100,
        0b01111110,
        0b11111111,
        0b11111111,
        0b10001001,
        0b00001000,
        0b00101000,
        0b00111000,
    ),
    '10': (
        0b00101010,
        0b00011100,
        0b01111111,
        0b00011100,
        0b01111010,
        0b11111100,
        0b00010000,
        0b00110000,
    ),
    '11': (
        0b00001111,
        0b00011110,
        0b00111100,
        0b01111111,
        0b11111110,
        0b00000100,
        0b00001000,
        0b00010000,
    ),
    '13': (
        0b01010010,
        0b11010011,
        0b00111100,
        0b00100111,
        0b11100100,
        0b00111100,
        0b11001011,
        0b01001010,
    ),
    '50': (
        0b11111111,
        0b00000000,
        0b11111111,
        0b00000000,
        0b11111111,
        0b00000000,
        0b11111111,
        0b00000000,
    ),
}
DIGITS = {
    ' ': (
        0b0000,
        0b0000,
        0b0000,
        0b0000,
        0b0000,
    ),
    '0': (
        0b0111,
        0b0101,
        0b0101,
        0b0101,
        0b0111,
    ),
    '1': (
        0b0010,
        0b0010,
        0b0010,
        0b0010,
        0b0010,
    ),
    '2': (
        0b0111,
        0b0001,
        0b0111,
        0b0100,
        0b0111,
    ),
    '3': (
        0b0111,
        0b0001,
        0b0111,
        0b0001,
        0b0111,
    ),
    '4': (
        0b0101,
        0b0101,
        0b0111,
        0b0001,
        0b0001,
    ),
    '5': (
        0b0111,
        0b0100,
        0b0111,
        0b0001,
        0b0111,
    ),
    '6': (
        0b0111,
        0b0100,
        0b0111,
        0b0101,
        0b0111,
    ),
    '7': (
        0b0111,
        0b0001,
        0b0001,
        0b0001,
        0b0001,
    ),
    '8': (
        0b0111,
        0b0101,
        0b0111,
        0b0101,
        0b0111,
    ),
    '9': (
        0b0111,
        0b0101,
        0b0111,
        0b0001,
        0b0001,
    ),
    ':': (
        0b0000,
        0b0001,
        0b0000,
        0b0001,
        0b0000,
    ),
    '.': (
        0b0000,
        0b0000,
        0b0000,
        0b0000,
        0b0001,
    ),
    '-': (
        0b0000,
        0b0000,
        0b0001,
        0b0000,
        0b0000,
    ),
}

matrix = Matrix8x8(I2C(-1, Pin(5), Pin(4)))
wait = 0


def show(image, dx=0, dy=0):
    for y, row in enumerate(image):
        bit = 1
        for x in range(8):
            matrix.pixel(dx + x, dy + y, row & bit)
            bit <<= 1
    matrix.show()


def show_text(text):
    x = 0
    for c in reversed(text):
        if c in '.:-':
            show(DIGITS.get(c, DIGITS[' ']), x - 1, 1)
        else:
            show(DIGITS.get(c, DIGITS[' ']), x, 1)
            x += 4


def get_weather():
    r = urequests.get("http://api.openweathermap.org/data/2.5/weather"
                      "?q=%s&appid=%s" % (CITY, API_KEY)).json()
    return r["weather"][0]["icon"], int(r["main"]["temp"] - 273.15)


while True:
    if wait <= 0:
        icon, temp = get_weather()
        wait = 60
    if icon[-1] == 'd':
        matrix.brightness(9)
    else:
        matrix.brightness(2)
    matrix.fill(0)
    show_text('%2d' % temp)
    utime.sleep(4)
    show(ICONS[icon[:2]])
    utime.sleep(4)
    wait -= 1