If an entire blizzard load of super cold rocks hit the Earth at the same time, the Earth's problem won't be with dealing with any cold it will be in dealing with all the dust raised into the atmosphere and the seismic shocks created by the rocks impacting the ground.

Instead of pretending that the earth is a lone ice box floating in space, and calculating what happens when we drop a large ice cube into it, I think it's more useful to simply ask: "How quickly will the sun melt our ice cube?"

The sun will contribute enough energy to warm the asteroid by 1 K every 6s or so, for a small asteroid. To raise the asteroid from 3 K to earth mean surface temperature of 288 K will take somewhat less than 1800 s, or about 30 minutes. That's about how long an ice cube would last in a chilled beverage on a not particularly warm day.

Now, that is roughly equivalent to turning off the sun for 30 minutes. That would have a noticeable effect on global weather, but I doubt it would remain noticeable even a few days after. Over the course of a year, this blip would amount to a solar energy reduction of 0.005%. Hardly enough to detect in the climate record.

Of course, we picked the smallest asteroid for the initial calculation. If you go towards the large end, you can crank up all these numbers by a factor of a million, more or less. Ceres, for instance, would suck up about 5000% of the sun's annual energy sent to earth. That's 50 years of solar energy, and obviously a significant impact. Given that it's round, we can debate whether Ceres counts as an "asteroid" or not. Nysa is more "asteroid-shaped", but only has about 100x the heat capacity of our initial calculations. So, a solar dip of 0.5% over the course of a year might be enough to notice, but likely not enough to cause attributable long-term climate effects (given that the Milankovitch cycles produce larger variance).

Note that if you start with a more typical asteroid temperature of about 200 K, you would reduce all the energy budget numbers to about 1/3 of their value (meaning, the smallest ice cube would melt in ~10 minutes).

$$m \cdot \Delta t \cdot c = m\, \mathrm{kg} \cdot 285\, \mathrm{K} \cdot c\,\mathrm{J}\,\mathrm{kg}^{-1}\mathrm{K}^{-1}$$

Since the escape velocity of Earth is 11 km/s (11000 m/s) the total energy coming in (in kg m^2/s^2 aka Joules) will be:

$$\frac{1}{2}mv^2 = \frac{1}{2} m \mathrm{kg} \cdot 11000^2 \,\mathrm{m^2}\mathrm{s^{-2}}$$

We want our asteroid to be capable of absorbing more energy than the energy it has from falling:

$$m \mathrm{kg} \cdot 285 \mathrm{K} \cdot c\mathrm{J}\,\mathrm{kg}^{-1}\mathrm{K}^{-1} > \frac{1}{2} m \mathrm{kg} \cdot 11000^2 \,\mathrm{m^2}\mathrm{s^{-2}}$$

Mass cancels out immediately and after some arithmetic, we end up with (rounding quite a bit) c > 200,000 J/kg K.

Unfortunately, the material with the highest specific heat is hydrogen, with a specific heat of about 14,000 J/kg K, which is nowhere near what we need.

Short answer: No.

Comparison: Canada has an area of 10 million km2. If you had 1/10 of a meter of ice (4 inches...) you would have 1/10,000 of a km * 10 million km = 1000 km3. So your comet has about the four times the chilling effect as spring in Canada. (Ok, 10 cm is arbitrary.)

Or the same as 40 cm of ice on Canada.

Now a square meter with 40 cm of ice would be about 400 kg. (Please, do NOT muddle the waters with the different density of water and ice. This is a BOTE calc.)

1 kg of ice takes about 500 kJ to melt, assuming it's starting from close to absolute zero. 400 kg takes 200,000 kJ

Solar constant is about 1 kW/m2 at noon. (Yes this is variable depending on latitude, time of year, cloud cover, and things that otherwise were thought only to affect the flavour of pizza) So 200,000 seconds of noonday sun would melt the ice. About 55 hours.

Which would be about 200 hours due to night, and low angles. 8 days. Ish.

However Canada isn't the whole world. Only about 1/50 of it. so 8/50 or about 4 hours sunlight over the earth to melt the comet.

Like throwing an ice cube from your drink into a campfire.

You're left with nuclear winter possibilities. Others have addressed that.