Not going to lie, this puppet is rather uniquely suited, lore-wise, for the Weeping Angels. Pretty damn slow, compared to most every powerful creature. Pretty damn fragile, compared to most every creature too.
But this puppet cannot blink, and can keep its head focused on weeping angels however it wants. It can also create more button eyes. If they stare into my eyes, then their infection will simply become another color added to me, as they are helping scribble on the tabula rasa that is the Monotone Puppet. Their infection will only benefit me.
Unfortunately, as of right now, lore-wise, I can't really... damage them. At all.
And they cannot displace me through time, because they are unable to absorb time energy from the force that animates me and keeps me in this reality. To give an example, I hug them at 1:30 pm at 12/25/2222 exactly. That means I stay at 1:30 pm at 12/25/2222 permanently until the Stars get bored of their creation being paused and pull me out of this boring predicament I've gotten myself. They promptly turn all of the Weeping Angels into a Star that is a FRIEND to all to prevent this glitch from ruining their entertainment again.
However, even if they could displace me...
I am immortal. So that just means I get to explore a new time period.