Does the elliptical orbit of the Earth have effect on Earth's climate?
Short answer.   When the Earth is closest to the sun in January that softens the seasonal variation in the Northern Hemisphere and it increases the seasonal variation in the Southern Hemisphere, so for now, in the Northern Hemisphere, the Summers are colder than they would be with a circular orbit and the winters are hotter.   The opposite is true in the Southern Hemisphere.

Overall the eccentricity variation doesn't change how much solar energy hits the Earth by much, so it's not the total solar energy.   It's when combined with precession, and the effect on the seasons, that the global effect can be significant. 

If we had a perfectly circular obit then the only Milankovich cycle that would matter would be axial tilt.  Ice ages would either be on a 41,000 year cycle - very neat and tidy, or they wouldn't happen at all cause the orbital variation wouldn't be enough to get ice ages started.  Other factors, like solar minimums could play a role and over longer periods of time, continental drift and variations in plant life, so, maybe not that exactly that, but I think it's likely.

Too long answer:   Hotter summers and colder winters (or the opposite) is a form of climate change that accompanies the orbital cycles.  All the orbital cycles change climate at least by seasonally.  But if I'm reading your question right, you're asking about global temperature, not just seasonal variation.

Eccentricity has an effect because the two hemispheres are quite different and because it amplifies the effect of precession.

Oceans tend to be a temperature buffer.  They circulate and they warm up slowly and they cool down slowly so they don't get too hot in summer or too cold in winter.  Land can vary in temperature significantly more.   This buffering can be seen in this average temperature chart of the two hemispheres.   



The Northern Hemisphere varies about 13 degrees C and the Southern about 5.5 C.   Those numbers might seem low, but remember that near the equator, seasonal variation is small and there's more land near 0 latitude than up near 90 or -90.   How accurate this chart is, isn't really relevant.   It doesn't need to be perfect.  What's relevant is that seasonal variation is much smaller in the Southern Hemisphere because so much of it is ocean, even with the Sun being closest in January which should make the Southern Hemisphere season variation more extreme.  The ocean is the key factor.

With that in mind we can ignore the smaller variation in the Southern Hemisphere and just look at the Northern Hemisphere variation for any orbataly driven climate effects.   Before we get into eccentricity, it's important to understand Axial Precession.   The Earth, as you noted, is closest to the Sun in early January, January 4th this year.  that 3.4% closer That means Earth gets nearly 7% more sunlight in the first week of January than it gets in the first week of June.   That 7% is too small to move the seasons, but it's enough to affect them a bit.   Earth overall gets more solar energy January (Northern Hemisphere Winter) and less solar energy in June.   The Northern Hemisphere's more exteme seasonal variaiton comes into play when the Earth is closest to the Sun in May-June-July-August - that should happen in about 8,000 years, give or take.  The cycle averages about every 23,000 years, so roughly 11,500 years between highs and lows for each hemisphere.   

The most relevant factor is the Northern Hemisphere Summer.   This is what is thought to drive ice ages back and forth.   Snow can last on land a long time in higher Northern latitudes and when the summers are less hot, the snow can last well into May, June, even all year around.   The warmer winters are less of a factor cause you still get snow in warm winters, sometimes, it snows more heavily when the winter is warmer.    It's the cooler Northern Hemisphere Summers that allow snow build-up and make ice ages possible.  Eccentricity (combined with precession) plays a key role on the periodic, colder Northern Hemisphere summer.   

Don't worry, we're not entering an ice age in a few thousand years.   We're actually in a cool Northern Hemisphere summer period now, but, It's not quite that automatic.

The eccentricity in effect, determines how significant the axial precession is.   When the orbit is nearly circular, the difference in energy perihelion to aphelion is virtually zero.   Orbital eccentricity is currently dropping, moving towards circular, so we're not in a period of expected significant changes.  (The increased CO2 would prevent an ice age now anyway, but that's another story).  Earth's orbital cycles are in more of a holding pattern than moving towards an ice age. 

When eccentricity peaks, the variation in solar energy hitting the Earth, perihelion to aphelion is significant.   It's loosely speaking, about 4 times the eccentricity.  The current eccentricity is 0.017, the current variation, Early January to Early June is about 4 times that or 0.068 or 6.8%.   At peak eccentricity of about .0679 would work out to roughly 27% variation in solar energy hitting the Earth, Perihelion to Aphelion.  (Perhaps nearly 30% with the working out the square).   This 30% variation in solar energy, depending on the perihelion and aphelion fall on the calendar, is sufficient to make ice ages come and go.   ((For very high eccentricity this 4 times rule doesn't work, but it applies pretty close to eccentricity pretty close to 0.))

In laymans terms, think of the eccentricity as turning up or down the effect of axial precession and those two combined, when properly lined up with the hemispheres, have a significant effect.   

The 3 cycles together, make a kind of ugly chart of ups and downs.  The orbital effect is rather uneven but Ice ages are often periodic because of feedback mechanisms on Earth.  Once formed, Glaciers are hard to melt and once melted, it takes peak conditions to get them forming again, so many of the smaller ups and downs in the chart don't drive ice ages, only the extremes.  

Wikipedia has a nice graph on the Northern Hemisphere solar energy at summer solstice.   When the chart is high, that tends to be a period of glacial melt.  When it's low, that's a period of ice age formation.   

Because the eccentricity is currently kind of low and shrinking, we're probably not due for an ice-age for about 50,000 years, at least according to current orbital models, per Wikipedia (same link as just above).