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.
The beginning of the year (1st of January) has been chosen for historical reasons (near the winter solstice) and has absolutely nothing to do with Earth's perihelion passage. Earth's perihelion could as well have occurred on the 10th of October. That it is near the 1st of January is pure chance (see for instance a post by EarthSky).
It is, therefore, not a discrepancy that needs to be corrected.
The only known relationship between the epoch of the Julian calendar (i.e., January 1st, 45 BC) and any significant solar moment was its synchronization with the first new moon following the previous year's winter solstice :
It was probably the original intention of Caesar to commence the year with the shortest day. The winter solstice at Rome, in the year 46 B.C., occurred on the 24th of December of the Julian calendar. His motive for delaying the commencement for seven days longer, instead of taking the following day, was probably the desire to gratify the superstition of the Romans, by causing the first year of the reformed calendar to fall on the day of the new moon. Accordingly, it is found that the mean new moon occurred at Rome on the 1st of January, 45 B.C., at 6h. 16′ P. M. In this way alone can be explained the phrase used by Macrobius: Annum civilem Caesar, habitis ad lunam dimensionibus constitutum, edicto palam proposito publicavit. This edict is also mentioned by Plutarch where he gives the anecdote of Cicero, who, on being told by some one that the constellation Lyra would rise the next morning, observed, Yes, no doubt, in obedience to the edict.