2010 Prius engine noise on startup (cold day)

How many miles on her?

 

Is the cat still there?

have you cleaned the egr circuit?

are you losing any coolant?

have you followed the maintenance schedule?

 

The EGR is Exhaust Gas Recirculation, a system comprising two components bolted on at the top/left corner of the engine. It takes exhaust gas from the exhaust manifold, cools it somewhat, runs it through the intake manifold where it goes back into the combustion chambers, under certain load conditions.

It was introduced with 3rd gen, is prone to clogging with carbon, and head gasket failures are a common occurence when it's clogged.

That knock at start-up is often due to coolant leaking past the head gasket into the cylinders.

The start-up knock can also be caused by running the engine cold for just a few seconds, then shutting down at letting it sit. Knock happens at next start up

Any warning lights or codes? Is coolant level in the engine coolant reservoir stable?

113K is a good time to get on the EGR cleaning, including the intake manifold. Consider an oil catch can too.

See first two links in my signature.

 

If it knocks I’d shut down. If it doesn’t, odds are better it’s due to that momentary start up.

still, EGR cleaning is good to to do ASAP.

Again, is coolant level in reservoir stable? Either way, keep an eye on it.

head gasket integrity is commonly tested with boroscope inspection of cylinders with coolant system pressurized (looking for coolant trickle), and/or leak-down test, where cylinders are pressurized one at a time, with special gauge set, with piston at TDC of compression stroke. Mechanics should know those.