When is Earth closest to the Sun? When is Earth closest to the Sun?

Well, you say, perihelion of course.  But perihelion is when the Earth-Moon barycentre is closest to the Sun. Not Earth itself.

The barycentre is on average 4,671 km from the Earth, which should mean that even if the orbit of the Earth-Moon barycentre around the Sun were perfectly circular, the Earth-Moon orbit around the barycentre would cause an oscillation in the Earth-Sun distance by twice that amount on a monthly basis.  That should be enough to move Earth's closest approach to the Sun away from perihelion.

Considering this, how many days from perihelion may Earth's closest approach to the Sun be?

Question inspired by: What place on Earth is closest to the Sun?
Earth's perihelion timings and distances are based on the distance between the center of the Earth and the center of the Sun. They are effectively the times and distances of the closest approach between the two bodies, not the barycenters. As we can see on Fred Espenak's astropixel page about perihelion and aphelion, the time intervals between two successive perihelions tend to vary between 363 and 368 days. 

The data on astropixels seems perfectly valid. With JPL Horizons, for example, the next perihelion is calculated to be on 2020 Jan 5 at 07:47:42, at a distance of 0.98324356482 AU (147,091,144 km) which matches astropixel's values. 

I analyzed the perihelions of the 21st century to see how much the actual timing can differ from an idealized "average" perihelion moment (based on the anomalistic year length). Perihelions this century remain within 1.4 days of that average. The differences from the average are mainly due to the gravitational effect of the Moon.

Using all perihelion times in the 21st century, and the average perihelion interval (anomalistic year, 365.2596 days) I constructed the following graph showing these differences: