Your "without correction" method isn't actually correct. If you were to go without a correction you would just look at the area greater than or equal to 196 (exactly what the probability statement says).

So the correction is helping.Code:`> 1-ppois(195,168) #exact [1] 0.01876504 > 1-pnorm((195.5-168)/sqrt(168)) #with correction [1] 0.01693269 > 1-pnorm((196-168)/sqrt(168)) #without correction [1] 0.01537678`