Based on h_t_t_p_s://www.home-assistant.io/docs/installation/raspberry-pi/ (my instalation) and some google search, i,ve update python 3.7.0 to 3.8.6 with no major problems, Just had to adjust some configuration in configuration.yaml, and just work like charm, and test in other instalation version 3.9.0 and run as well too, Hope this help:
#dont have to change anything in home assistant “boot”.
#With PI user
sudo apt-get update
sudo apt-get upgrade -y
sudo apt-get install build-essential tk-dev libncurses5-dev libncursesw5-dev libreadline6-dev libdb5.3-dev libgdbm-dev libsqlite3-dev libssl-dev libbz2-dev libexpat1-dev liblzma-dev zlib1g-dev python3 python3-dev python3-venv python3-pip libffi-dev libtiff-dev autoconf libopenjp2-7
wget -O /tmp/Python-3.8.6.tar.xz https://www.python.org/ftp/python/3.8.6/Python-3.8.6.tar.xz
cd /tmp
tar xf Python-3.8.6.tar.xz
cd Python-3.8.6
./configure
sudo make altinstall
sudo apt -y autoremove
cd
rm -r /tmp/Python-3.8.6
rm /tmp/Python-3.8.6.tar.xz
sudo mv /srv/homeassistant /srv/homeassistant_old
cd /srv
sudo mkdir homeassistant
sudo chown homeassistant:homeassistant homeassistant
sudo -u homeassistant -H -s
cd /srv/homeassistant
/usr/local/bin/python3.8 -m venv .
source bin/activate
python3 -m pip install wheel
pip3 install homeassistant
pip3 install websocket-client #sonoff if add in costum components
pip3 install mysqlclient #mysql if run HA db in mariadb/mysql
hass # here we see the error´s and adjust config
for version 3.9.0 just change “3.8.6” to 3.9.0 and
“/usr/local/bin/python3.9 -m venv .”
#Best regards Igor