Before deciding that the area stays wet for longer than a couple of days, be aware of what it does during the typical growing season. Right now I have ponds in the well-drained farm field next to my house because the top inches are frozen and so won’t drain. Often living in a new place for a year before making major changes to the plantings is helpful so that you see the whole cycle of what is already there and how sun, shade, and moisture vary over the course of the year. Was last year wetter than usual or typical of your area based on long term weather comparisons?

Regardless, there are a reasonable number of plants that tolerate both wet and dry settings. In my area, New England, two that do that are Ilex verticillata aka winterberry holly, a deciduous native holly that lights up the fall and early winter landscape with bright red until birds eat the berries, and Clethra alnifolia aka summer sweet, a shrub with late summer flowers and glowing yellow autumn color. Others include Itea viginica aka Virginia sweet spire, with late summer flowers and bright red fall color, and red twig dogwood aka Cornus sericea.

You will want to see how much light the area gets and make a measured plan with what is already there that you want to keep. (A printout of satellite imagery such as Google can be helpful with this.)

It looks like you might have some red maples, and IME it can be difficult to get much to grow well near established maples due to root competition and shade, so what is already there can have a strong influence on the plans that can be successfully implemented.

Can you give us your nearest big city? What will grow in different parts of the world, even in zone 6, can vary considerably.

The ton of vines that fill in for complete privacy in the time of vegetation and what looks like a row of woody trees and shrubs embedded in the chain link fence already are going to be a clue as to what will grow in this puddle. It's just a matter of choosing the right plant for your location and you can fill it in with plants.

Keeping a dog, puppy or not, away from the fence perimeter, puddle or not, is another department. Plants don't often work.

I'll toss in Leucothoe, common name Dog Hobble to NHBabs list of good possibilities.

Thank you both for the great suggestions. @NHBabs z4b-5a NH, I live near Detroit, MI. I think last year's weather was pretty typical for the area. You make a good a point regarding the frozen soil and now that I think about it more, the moisture was most problematic around winter and early spring. In the summer it didn't tend to pool for nearly as long. @Christopher CNC, also great point about the vegetation on the chain. In the summer, the area filled in with a lot of offshoots from the woody shrubs, along with weeds, queen anne's lace and whatnot. It does get a little shady right along the fence, but the fence is south facing so we get a lot of sun overhead.

You could plant a big grouping of hostas. They don't mind shade, seem to like soil on the moist side. Hostas can also use some of the water. Having a full garden of hostas may keep dog away from puddle, but they don't start growing until spring, probably after the puddle goes away.

You probably have heavy clay soil so water doesn't soak in so well. My parents get a puddle in the back of their yard every spring that goes away after about 3 weeks. You could dig out some of the soil and replace it with soil that would absorb the water. But who knows how deep the clay goes or how much you may have to dig out to achieve that.

Thanks, @Izzy Mn! We love hostas. I separated and transplanted several in our front yard last year and we plan to add more. Could be nice and cohesive to add more in the back. We’ll have to do the hole test to see how long it takes for water to disperse and what kind of soil we’re looking at.