Turn on light at different times depending on Day

Okay, here you go, it’s just like your first code but it has a delay with a template in it.
Basically it fires at 05:25 and if it’s a weekday (0=mon, 1, 2, 3, 4=fri) it will set the delay to 0 if it’s NOT mon-fri it will set the delay to 55 65 minutes (just as you asked)

You will need to copy and paste this in ‘Over’ your old automation and do it via an editor (notepad, atom etc. This assumes you have samba or can use ssh to then use nano)

- id: TurnOnTPL
  alias: 'Turn On TPL'
  trigger:
    - platform: time
      at: '05:25:00'
  action:
    - delay: >
        {% set offset = '00' if now().weekday() in (0,1,2,3,4) else '65' %}
        "{{ '00:' ~ offset ~':00' }}"
    - service: homeassistant.turn_on
      entity_id: group.tpl

Note that your old indentation was ‘slightly’ wrong (all spacing should be in multiples of two)
Also I’ve made your trigger and actions into lists (so you can add more (worry about that later))

I edited the above as the difference wasn’t 55 but 65 (yes, Ive had 300 minutes in these slots before now)