Barycenter? Not!
Some claim that the Earth's insolation is a function of its distance from the Sun (it is) and that this varies because the Earth orbits around the Solar System's "barycenter" (it does not). That is nonsense, perhaps outright dishonesty, especially if you hear it from a climate denier who claims to know physics (wrong three ways).
The Truth
(as we understand and measure it)
The gravitational acceleration a_A of object A towards object B is proportional to the mass of object B ( m_B ) divided by the square of the distance between them ( r_{AB} ).
F = { \Large { { G m_B m_A } \over { r_{AB}^2 } } } ~ ~ ~ and so ~ ~ ~ a_A = { \Large { F \over m_A } } = { \Large { { G m_B } \over { r_{AB}^2 } } } ~ ~ ~ and also ~ ~ ~ a_B = { \Large { F \over m_B } } = { \Large { { G m_A } \over { r_{AB}^2 } } }
There are some tiny modifications described by general relativity, but those second order effects only become measurable when distances are small and masses are enormous.
To be pedantic  the forces F on object A and object B are equal (though opposite) vectors pointing at each other. The equations above would be more accurate if expressed as vectors, but less understandable for most readers. If you are interviewing for an orbit designer job at NASA, and they ask you this, ask them whether they want the relativistic equations, the vector equations, or the simpler equations above. When NASA estimates mission costs, they use vectors, and when they compute precise space probe trajectories, they use general relativity.
When NASA delivers a lander to Mars, it must enter the Martian atmosphere with kilometer precision after a 500 million kilometer journey from Earth. They need 9 decimal place accuracy, and include gravitational effects from all the inner planets in those calculations.
However, if we merely wish to compute the distance from the Earth to the Sun to estimate solar insolation, we can pretty much ignore the other planets, because they are far away, and accelerate both the Earth and the Sun almost equally, just as the Earth pulls on both you and the vehicle you ride in almost equally.
The difference between almost and zero occurs when two objects are at slightly different distances from the object that is pulling on them. The Earth is on average about 1 astronomical unit (au) from the Sun, 149,597,870.7 kilometers. Earth's orbit is actually slightly elliptical; I'll touch on that later.
Note the ten decimal place accuracy for the definition of the astronomical unit; with communication links to space probes around the solar system, and the speed of light, we can measure distances to space probes billions of kilometers away with centimeter accuracy.
Jupiter is the second largest mass (that we know about!) in the solar system; the giant planet is 1/1047 of the mass of the Sun. Jupiter orbits 5.2 au from the Sun on average. Jupiter's orbit is more elliptical (4.9%) than the Earth's (1.67%), but let's call it a circle for now.
As Jupiter slowly orbits around the Sun, it moves closer (4.2 au at opposition) and farther (6.2 au at conjunction) from the Earth. Jupiter pulls on both the Earth and the Sun (and vice versa!), but accelerates the Earth 53% more at opposition and 30% less at conjunction than it accelerates the Sun. While that seems like a lot, remember that the Sun is on average 5.2 times closer and 1047 times more massive, so these accelerations are only 54 ppm (parts per million) and 24 ppm of the Sun's gravitational acceleration on the Earth (about 6 mm/s²). In both cases they accelerate the Sun and the Earth apart; they behave like lunar and solar tides do on Earth/ocean spacing. Acceleration, not spacing. This will displace the Earth's orbit a few ppm towards Jupiter. Jupiter is a moving target, with an 11.8 (earthyear) period; that will tend to reduce the displacement effect, likely less than 20 ppm on average.
I'll put a real calculation in here someday.
Compare 20 ppm annual spacing variation to 33400 ppm annual spacing variation due to Earth's "small" eccentricity, and you can see why Jupiter has no effect on short term climate variation.
In the long term, tens of thousands of years, those tiny annual nudges add up to radically change the eccentricity of the Earth's orbit, while solar tides turn the Earth's 23 degree axial tilt around the sky (precession of the equinox). These long term effects add up to the Milankovich cycles. Those, which change the annual heat flux to the Antarctic ice sheet versus the Artic ocean, and are responsible for the ice ages. They have no discernable short term effect on global temperature and climate.
Here's a table of perihelion and aphelion distances for the 21st century, which show how small the orbit changes are:
http://www.astropixels.com/ephemeris/perap2001.html
If you disagree, please do not work for any national space program.
The Solar System Barycenter
and why it is meaningless
The barycenter of the solar system is the hypothetical center of mass of the solar system. That notional point is what (approximately) orbits the Milky Way galaxy, but it has no effect on the orbital distances of the planets around the Sun, which are determined by their physical distances from each other, not to some arbitrary point in space. Space attracts nothing.
Planet 9 is a hypothetical outer planet 400 to 800 a.u. from the Sun, 5 to 10 Earth masses, inferred from very weak gravitational effects on Pluto and other dwarf planets. Let's call it 600 a.u. and 7.5 earth Masses, or 600 Moon masses.
We cannot even measure where the hypothetical barycenter is. Imagine a simplified universe comprising only the Sun, the Earth, Jupiter, Planet 9, and a Moonsized mass (the "Zoon") 10 billion light years away (632 trillion astronomical units). The relative masses (in Zoons) and effects on the barycenter:

Mass 
Distance 
effect 

zoons 
a.u. 
(relative) 
Sun 
2.7e7 
1 
2.7e7 
Jupiter 
2.6e4 
5.2 
1.3e5 
Earth 
81.3 
1 
81 
Planet 9 
600 
600 
3.6e5 
Zoon 
1 
6.3e12 
6.3e12 
If barycenter mattered, then Planet 9 would have almost three times the effect of Jupiter, and Zoon would 2.3 million times the effect. The actual mass out there is more than 1e23 Suns, 3e30 zoons, and the average distance is more than 10 billion light years.
Yet there are still science deniers who yammer about some hypothetical barycenter as if they know what that is, where that is, or how to calculate it, and that somehow the Earth orbits around this barycenter once per year, with strong effects on solar illumination and climate. It is unwise to employ such people in professions requiring knowledge of counting, such as bus driver or convenience store clerk. They must be disregarded regards climate policies. Not ignored; rattlesnakes can't do these calculations either, but they can bite.