1.  You should NOT discourage your partner from having a relationship with his 9 year old son.. 

2.  Your kid isn't perfect and I'm sure your partner deals with plenty from your son too... believe me.. your kid doesn't walk on water.. no kids do.

3.  You do have a say in how your household is run.. ground rules for the kids and you and your SO need to mutually agree on these.. but it is up to each of you to parent your children...  

4.  Your DH may not get much time with his kid.. so may not want to spend all that time fighting.. but maybe decide what are some things that are positive changes that he can start on.. it won't all happen at once.. work on one thing and then move to the next.

5.  If you cannot be around his kid.. you should think about doing other things on his weekend so you aren't around as much.. or maybe not be in the relationship.

6.  your SO will come to resent you if you keep this up.

What is the visitation schedule?

Paste the schedule on the wall and tolerate no additional visits.  If DH wants to decline a visitation, fine. But never tolerate extra visits.  If BM fails to deliver the Skid per the schedule, ask DH why he isn't nailing BM with a contempt motion for withholding SS per the schedule.

We were the CP side of the blended family equation in our world.  We never allowed deviation from the schedule.  If they chose to forgo a visitation, fine. But if they did not put him on the plane per the end of the visitation, it was game on.

As for responding to his demands.. NO! solves that problem.  Visiting kids immediately engage in the household following the rules, complying with the standards of behavior and standards of performance... or.... they learn abject misery inducing escalating consequences.

You are not rejecting his sons while DH accepted yours.  You are requiring compliance to the visitation schedule.

Do not demonize yourself and do not allow drop in unstructured Skid visits.  Of course the same should apply for your son as well.

IMHO of course.

Take care of you.

She's a girl but is very easy going and independent.  She's always on her computer, reading books, playing with toys, playing with friends, painting, etc.  We do stuff together but she never comes to me cause she is bored or needs constant supervision.  That being said she has some fellow 9 year old friends who are not independent and it was very eye opening to me how blessed I am.  This friend of hers was constantly coming in my bedroom, "Can I have a snack?"  "Can we go to the park?"  "can you take us to the skating rink?"  We have a new $2000 bengal kitten and this friend is just oblivious.  I caught her leaving the back door wide open.  She would take out every single one of our board games and scatter all the pieces and not put anything away.  She will eat all the snacks and lunch box food I have budgeted for two weeks in a few hours if you aren't watching her like a hawk.  So I get it.  I had to sit down with the 9 year old best friend ( and her mom is my good friend) and say "look here.  You can't leave the back door open and let out my brand new kitten.  Absolutely not.  You won't be invited back.  You can't get all of Khloe's food for her lunches.  Here are my rules.  If you break my rules, you won't be back."  and I guess in your case your SO needs to get on the same page regarding ground rules.  I had to talk with my friend. I took them to the movies and her kid was up running around the theater and leaving the theater to roam the halls.  I said, "I"m not taking your kid to the movies anymore unless she knows the groundrules."  I don't know how my friend copes honestly.  If that were my kid and I couldn't escape.... OMG.  So I feel your frustration.  I just get a blast of an unruly 9 year old every once in awhile.  It's horrible.         

My wife was like this with our SS who is now 9. Maybe not quite as bad.  I was able to change things by making her see that indulging him is bad for him.  In 10 years, he is supposed to be an adult.  My SS is slowly getting there but would be 5 at home without me.   Your husband needs to discipline him.  His son needs a father not a buddy.  I teach middle school and so many teachers want to be their friends.  I am firm but fair and kids like my better than in my first couple of years when I was lenient.  Kids actually crave structure and discipline.  If you can get your husband to change his mindset, it will be better for all.  I would talk to him about a list if expectations before he arrives.  They should then be posted in clear sight.  If he tries to fight it, you step in.

SS is there to see his father. So his father needs to spend time with his son and take care of his needs. Can you get away for the weekend?

I'm am one of the few stepparents who is not a fan of telling the Non Custodial Parent (NCP) to adhere strictly to the visitation. Your partner obviously did not plan on being a part-time father. It would be terrible of you to discourage him from seeing his son even less. How awful. 

There is always a possibility that your partner will have SS 100% of the time. ALWAYS. BM could pass away or become mentally or physically incapacitated. Would you stay if that hsppened?

Cleaning him cleaning up after him.  If there is nothing for breakfast, DH is going out and getting food. DH Should make sure there's food for his DS. and food for the rest of the family.   If SS wast ice cream. DH Ggoes out and replace it.  When DH gets tired of going out   Things will change 

Your SS needs him to step up and PARENT his child.