Congrats on your new place! That patio area has lots of possibilities. I would first work on the gutter downspout by adding a 12 inch piece and then one of those extendable tubes that bend so you can have the water go toward the fence and under it.
Then get outdoor cleaner with bleach to clean up cement and walk path. If budget allows get wider pavers for the walkway. Kill the weeds and add pea stone around the entire area. Get some different height flower pots to have colors you like and some greenery. You can probably find small pot holders for the fence too. Will be charming. Good luck to you!

I don't understand how the area with the table relates to the other one with your picture with the dog. Can you draw a plan of the area to show where one is in relation to the other?

IMO thta silly little concrete slab is not ideal You could do a free standing deck to cover that whole area with weedfabric underneath . You will have to take the dog for a walk a few times aday since that smell will never go away unless you prep the soil and do gravel 4" deep that I find seems to do the trick in my dog run to avoid the urine smell but we do not eat out there .Taht fence could use a power wash too. I assume there are rules as to what you can do that is why I suggest a floating deck. The pea gravel is a good plan but it needs to be quite deep to control the smell and you need to be up on the pick up ofpoop no matter what.

@lisedv I pasted the two photos together so you can see how they are connected side by side based on the small porch and railing. Dog is standing on the small porch. You always have good ideas, so hope this helps!

You need some plants. Greenery will transform the space. If your dog is going in the same spot all the time, it will smell and probably kill any greenery. If you don't want your outdoor space to become a dog latrine, you have to take your dog elsewhere.

The Brits do small gardens well. I'd train some climbers up your fence. I'd plant a few bigger flowers in your corners and very short flowers in the narrower sections. Just putting a bunch of annual impatiens would do a lot for that space.

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If you look closely, you can see the railing/steps so it appears the table is to the right and the AC to the left of the steps.

I agree with stones and/or more pavers and containers for plants. I would pressure clean the slab and you can buy special paint for cement to improve the look.

Suggestions:

Budget friendly improvements: Refinish the fence.. Powewash the slab then put an outdoor rug over it. Add large potted plants. Don't use the back patio for a doggy bathroom.

Longterm: Build a floating deck over the slab to fill the space. Add large potted plants. This is Instadeck from Wayfair.

If weeds are going to be a constant battle, the last thing you want to deal with is gravel/stones. I suspect you are in a much wetter place than most of the respondents are used to. Knowing where that is will help with suggestions.

The downspout is going to be an ongoing issue. Talk to the HOA, or whoever controls outside your fence about where you can run it to. The slab cut off the obvious path.

Plant go in the ground. Pots are a hassle that isn't necessary.

This is in Fairfax virginia

nothing crazy wet but rain makes this area very soggy muddy

I would build a lattice cover for the a/c. Get an outdoor rug for the patio. Potted plants on the ground and hanging on the fence. If you have an outdoor faucet, a drip system works great. We got a Raindrip brand system from walmart that comes with a timer and plenty of 1/4' hose.

I wouldn't worry about planting in the ground. And no poisonous weed spray with pets.

You need a wider front walk with solid pavers, not stepping stones. Consider paving the yard with brick.

For one part of the yard, shredded cedar bark mulch works great, stays in place and safe for dogs. Our dogs love it. Has to be mulch from the soft bark, not wood chips. Or would spread that everywhere instead of brick.

I would add a rug under the table. Add artificial grass to the ground and potted plants along the fence.

Here's a budget friendly visual of my suggestions. I'm showing white furniture (inexpensive on Amazon) because the dark furniture you show doesn't stand out with the dark fence surround.

Another vote for decking the entire space. You need all the room you have. The way to get the max out of what's there is if the entire square footage is the same level and material.

A rug and grass’ll do it for me. 💚

Treat the patio with an algaecide and power wash. But, due to the close quarters, there is little air circulation and the algae will be an ongoing battle. I love the look of an outdoor area rug, but the algae may negatively impact it. I had one, but finally got rid of it for that very reason. If you decide to fill in the ground areas with pea gravel, be sure to put down a double layer of weed block first to control weeds and to keep the gravel from sinking and merging with the soil. I would add color by planting in containers and fence-hung baskets. Pay attention to how much sun/shade the area receives before choosing your plants. Enjoy your new space!

Could be a great space. I know it’s small, so using the vertical areas of the fence would afford you a garden without loosing square footage. Have a search for vertical garden systems. Agree with more substantive walkway pavers would make the entry more appealing. Regarding the odour issue, try a good spraying with household white vinegar - it works on cat urine odours so it wouldn’t hurt to give it a try. There might be an indoor pet tray system that could be affective outside that you could wash down with vinegar to keep the dog using one area so you don’t have to do a whole patio wash.

Walk your dog. I’ve had dogs and small yards and yes it is 100 times easier to open the door for a quick pee. BUT this is a tiny space you want to use for eatting in. No. Just break the habit. This needs to either be a dog bathroom or your patio. You decide.

Get quotes for paving the entire space. Then add pots with tall flowering plants. You don’t have to squish your table in the corner nor work about someone’s chair falling off the cliff.

If your ground is soggy now, just wait for mosquito season. You may not even want to be out here!

Someone suggested bleach. NEVER USE BLEACH ON URINE! Urine contains amonia and the combination will burn your lungs.

We had a very wet and muddy small, partially enclosed back yard. Rather than work french drains and hassel, we built up. We covered thw hole muddy mess with a deck. The mud is still there, but the deck hides it.

great ideas already - if it were me, I would want this spot to be clean and comfy for me and my dog so would probably start pretty basic

i like @tracefloyd suggestion of shredded mulch, we get hardwood mulch put in our beds and it really stays put, it’s soft and my Labrador will lay in it no problem

I also like the idea of 1 or 2 pretty pots to start with - you can see how the light/sun works this summer and make adjustments going forward

have some Simple Green/ pet odor eliminator in a large watering can and pour that on your dog’s spot after he goes

we have a oscillating outdoor standing fan from Decobreeze and it is such an easy way to keep mosquitoes away and cool down an area

If you start walking your dog then you'd be able to install pavers to make it into a large patio that's levelled with your cement slab. It's not a very big yard so I suggest you look into pricing it.

You can also build a wood/composite deck over the cement slab and vent.

If you want to invest, consider removing the slab. Make the whole area brick sized colored pavers. Include the steps and entrance porch.