The Zeiss cleaner should be fine (my order is blower then LensPen then Zeiss cleaner, although most of my lenses have a B+W MRC UV on them).
But those spots in your image aren't from the front of the lens - spots on the front element don't usually appear in images at all (unless the spots are penny-sized - and even then, they would not be in focus). So...in addition to spots on your lens, you've got dust on your sensor.
Agree, if you can see spots on the Lens Front Element, as advised try first with a LensPen, then Zeiss Cleaning Solution, if that doesn't get it off, back to Canon, you may have splashed something acidic onto the front element that's pitted the Lens Coating.
And it does look more like you have Dust on the Sensor, point the Camera at a light blue sky, slightly OOF, take a shot then check in your software, if you have the spots on the Image it's more likely a Dirty Sensor, Use a Rocket Blower, or better still The Arctic Butterfly Blower then try the Sensor Clean on the Camera a couple times, if that doesn't clear it, send to Canon for a clean.
A good piece of Kit is the Arctic Butterfly Brush, excellent for cleaning the Mirror & Sensor prior to getting out the Sensor Cleaning Swabs, again the "Visible Dust" work best for me.
I do my own Sensor Cleaning but quite honestly it's not for the faint hearted & unless you shoot in dusty environments (Like Africa), your better just sending it off to Canon for cleaning.