Why Earth's perihelion occurs 3rd January rather than 1st? Why Earth's perihelion occurs 3rd January rather than 1st January? Is there any effort to correct this discrepancy?
Back in 1246 CE, the perihelion coincided with the December solstice. The Earth was at perihelion at 1 Jan in 2005 and will be on 6 Jan in 2096, GMT time. The drift is not monotonic, but it does progress overall throughout the centuries. Eventually, it will drift towards the March equinox and beyond, to come back to the winter solstice in around twenty thousand more years. This is primarily due to Milanković cycles of Earth's eccentricity and precession index.

But the problem isn't just effectively unfixable; it's just not a problem in the first place. The apsides were seldom a concern for calendars. The practice of 1 Jan as the new year may have begun with the Romans, but it was not continuously practiced throughout Europe, e.g., the early medieval new year began at several different days from early to late March at different times in history, and sometimes at Christmas.