@benstear ,

If you want your workflow run weekly, you need to set Scheduled events for the workflow.
If you want the workflow runs only when have pushes in that week, you need to compare whether the commit SHA ( github.sha ) of current workflow run is different with that of the last workflow run.
You can use cache action to store and pass the commit SHA in each workflow run.

A simple demo:

on:
  schedule:
    - cron: '0 0 * * 0'
jobs:

test_schedule:

name: Test schedule

runs-on: ubuntu-latest

steps:

- name: Create cache file

run: |

mkdir check-SHA

echo ${{ github.sha }} > github-sha.txt
  - name: Check SHA
    id: check_sha
    uses: actions/cache@v1
    with:
      path: check-SHA
      key: check-SHA-${{ github.sha }}

  - name: Some subsequent steps
    if: steps.check_sha.outputs.cache-hit != 'true'
    run: echo "Exccute the subsequent steps"

You can see I set the key of the cache contains the  commit SHA  of current workflow run, when execute the cache action , it will check whether the specified key  matchs an existing cache. If the key  matchs an existing cache ( cache-hit is true ), it will restore this cache to the specified path on the local, and overwrite the file created in the " Create cache file" step in the path. If the key doesn’t match any existing cache ( cache-hit is false ), the action will save the file in the " Create cache file" step in the specified path  as a cache with the specified key.

In each subsequent step, use the if conditional to skip the subsequent step when the  cache-hit is true.