Thank you for your question. Of course it has an impact on mortality rates - as you correctly, state, there is an impact on \(P_x(t)\), which in turn impacts the exposure \(E_x^C\), which in turn impacts your mortality rate (since you use the exposure in calculating your rate).
You are also correct in stating that there will be more deaths after April.
But I suspect you also wish to know if there is an impact of this large influx of policyholders on the AGE to which the mortality rate (either your \(\mu\) or q) applies to. Recall that it is the RATE INTERVAL (RI) which allows us to attach ages to mortality rates, so we need to ask ourselves how this large influx affects the RI. Note also that the RI "does not really exist" in calendar time.
Suppose we consider a life-year rate interval. This large influx of new policyholders has absolutely no effect on the exact age at the start of the RI and the average age at death (which is the age in the middle), since the life-year RI cannot be "transposed" into calendar time. Note also that the assumptions used to derive the age at the beginning and age in the middle of the RI are not violated by the large influx of policyholders.
Suppose we consider a policy-year rate interval. Again, this large influx of new policyholders has absolutely no effect on the age at the start and the age in the middle of the RI. Again note that we cannot transpose this RI into calendar time. Also, we do not violate the assumption of deaths occurring half-way through the RI, since we cannot say anything on the relationship between the policy anniversary and 1 April. HOWEVER, if I told you that there was a large withdrawal (or alternatively an influx of) policies three-months after the policy anniversary, then there WOULD be an impact on the average age of death. Can you figure out why?
In addition, IF the RI was a calendar year RI, then the influx would have an effect on the average age at death. But we don't consider this case in our course.
The answer to your last question on lapsing at a particular time is precisely the reverse of what would happen with a large influx. No impact on the ages to which your estimates apply (there is no impact on the RIs and the assumptions we use to get the ages in the rate intervals), but there is an impact on the behaviour of the population.