Including cypress tests in CI

Yes, you can, and you can easily execute on CI with Cypress GitHub Action. You can refer to more details, via Cypress GitHub action repository https://github.com/cypress-io/github-action

Simple Example https://gist.github.com/NilukaSripalim/36e093530887ce6e5f5c889b2f2547e9

# CI name
name: Cypress GitHub Actions

# The on key is used to define when
# the CI should be triggered, aka Event
on:
  # When someone push or merge a pull request
  # inside the master branch
  push:
    branches:
      - master
  # When someone create a pull request from
  # the master branch
  pull_request:
    branches:
      - master

# A job is a set of steps that execute on the same runner.
# The jobs can be run in parallel or sequentially and can 
# have many steps.
# In this case, I created a single job with many steps
# since the configuration is pretty simple but
# I could have created a different structure using parallel jobs.
# Each job uses an own runner.
jobs:
  # Job name
  suite:
    # OS of the runner
    # # We're running on ubuntu-latest, nothing special
    runs-on: ubuntu-latest

    # A step is an individual task that can run commands in a job
    # and share the same runner.
    # Each step can run a ready-to-use action or a bash command
    steps:
      # Runner meaningful name
      # It's a step's name
      # As usual, we simply checkout the project
      - name: Checkout
        # In this line basically our workflow checks our repository 
        # allowing you to run actions against own code
        uses: actions/checkout@v2

        # This action is provided by Cypress.
      - name: Setup and run Cypress
        # This is a cypress ready-to-use action
        # that setups the runner installing the 
        # dependencies using NPM and installing Cypress
        # Execute all tests cases in project.
        uses: cypress-io/github-action@v2
        # The action accepts parameters.
        # You should read the documentation before using an action
        # to understand what it does under the hood.
        # Here Run the tests on Chrome & Headless Mode.
        with:
          browser: chrome
          headless: true
          record: true
        # after the test run completes
        # store videos and any screenshots
        # NOTE: screenshots will be generated only if E2E test failed
        # thus we store screenshots only on failures
        # Alternative: create and commit an empty cypress/screenshots folder
        # to always have something to upload  
      - uses: actions/upload-artifact@v1
        if: failure()
        with:
          name: cypress-screenshots
          path: cypress/screenshots
      - uses: actions/upload-artifact@v1
        if: always()
        with:
          name: cypress-videos
          path: cypress/videos