There is a reason for second collections.
Of the money donated to the general offertory collection, which is for parish operations, a percentage of that goes to the diocese for its operations. The diocese determines the percentage required from each parish. The wealthier parishes might be sending 10-15% to the diocese; the poorer parishes might be receiving a monthly stipend from the diocese, although I think this is rare.
But there are other collections for the Church on the parish, diocesan, national and universal levels, and no portion of these collections go to the diocese for its general operations fund. Some parishes rarely have second collections, because they send pre-printed collection envelopes to registered members of the parish, and/or places the special envelopes in the pews or near the entrance of the church.. So on some Sundays, you'll see parishioners placing two envelopes in the basket -- one is the general offering, and the other is for the special offering.
Some examples of special offerings would be USCCB's Catholic Relief Services (annual plus occasional for major catastrophes), USCCB's Catholic Campaign for Human Development (annual), the Vatican's Peter Pence collection (annual), the Holy Land (Good Friday), Black and Indian Missions (annual, since 1884 in the U.S.), diocese's Bishop's Appeal, Diocesan Seminarians, Priest and Religious Retirement (annual), etc. There might also be specific collections for parish projects, capital improvements, or for a parish's debt reduction drive. The diocese does not receive a portion of these offerings.
Specially pre-printed envelopes reduce the number of second collections during Mass, but parishioners frequently forget their envelopes, and many attendees don't sign up for the envelopes. It is up to the parish pastor to decide when to take up second collections.
There are two proper times to do the second collection -- immediately after the first collection, before the gifts are brought up to the altar (usually there is an announcement), or immediately after the Post-Communion prayer, during the announcements, prior to Dismissal.