Mongodb, can i trigger secondary replication only at the given time or manually?

Your idea of using a replica-set is flawed for several reasons.

First, a replica-set always replicates the whole mongod instance. You can't enable it for individual collections, and certainly not only for specific documents of a collection.

Second, deactivating replication and enabling it before you start your report generation is not a good idea either. When you enable replication, the new slave will not be immediately up-to-date. It will take a while until it has processed the changes since its last contact with the master. There is no way to tell how long this will take (you can check how far a secondary is behind the primary using rs.status() and comparing the secondaries optimeDate with its lastHeartbeat date).

But when you want to perform data-mining on a subset of your documents selected by timespan, there is another solution.

Transfer the documents you want to analyze to a new collection. You can do this with an aggregation pipeline consisting only of a $match which matches the documents from the last month followed by an $out. The out-operator specifies that the results of the aggregation are not sent to the application/shell, but instead written to a new collection (which is automatically emptied before this happens). You can then perform your reporting on the new collection without locking the actual one. It also has the advantage that you are now operating on a much smaller collection, so queries will be faster, especially those which can't use indexes. Also, your data won't change between your aggregations, so your reports won't have any inconsistencies between them due to data changing between them.

When you are certain that you will need a second server for report generation, you can still use replication and perform the aggregation on the secondary. However, I would really recommend you to build a proper replica-set (consisting of primary, secondary and an arbiter) and leave replication active at all times. Not only will that make sure that your data isn't outdated when you generate your reports, it also gives you the important benefit of automatic failover should your primary go down for some reason.