OK, I've attached some screenshots from your game with some red boxes to highlight key issues and things for you to pay attention to.
The first screenshot is from the Victory Progress screen, showing your level of influence over the other civs (ranging from a high of 94% with Indonesia and a low of 19% with Mayans). The Mayans are your biggest challenge.
The next screenshot is the Culture Victory tab from your Culture screen, which provides a rough overview of who has become influential over how many other civs (at this point, everyone is 0 of 7 civs), and the ideologies and happiness of the other civs. As you can see, Mayans are content, which is not helpful in the grand scheme of things.
The next screenshot is from the Influence by Player tab from your Culture screen. The important details here are your level of influence over each civ (Exotic, Friendly, Familiar, etc. -- the goal is at least Influential with each), the tourism modifiers that apply to your relationship towards that civ (you can mouse hover the percentage to see what modifiers apply), and how much tourism with that civ that modifier yields (i.e., 82 tourism x the modifier that civ provides (plus 100% of course) -- so India, where you have a +40% modifier, is experiencing 114 tourism from you).
The second Influence screen scrolls down to show your influence with the Mayans. It shows that you are merely exotic with them and have only a 6% modifier, resulting in only 86 tourism to them. The tooltip shows that you are suffering from a -33% tourism modifier due to different ideologies -- put a diplomat in Maya's capital to eliminate 25% of that -33% malus. If you can buy open borders from the Mayans (where they open their borders to you, but yours remain closed), you can pick up another +40% (since you've taken Cultural Exchange).
Notice that you can also flip the view on this screen to show influence from the standpoint of another civ. The last screenshot shows influence from the Mayan perspective. Note that they have the benefits of a +34% modifier for having more happiness than you (resulting from the Order tenet Dictator of the Proletariat). The only way to combat that is to boost your own happiness above the Mayan happiness.
As for what you are doing in your cities, you appear to be in good shape on theming bonuses and you are building a Hotel in Beijing, which will have a immediate positive impact. You are also building a Hotel in Nanjing, but that's a bit of a waste since you have no great works in that city. Your other two cities need Hotels much more urgently. You could use more artifacts, but I see you are digging one now and building another archaeologist, which is good.
As an aside, you built Hermitage in the wrong city. You should have built it in Beijing, where it's native bonus and the bonus from your religion would have the most impact (in fact, Beijing is the only city that still has your religion, so you aren't even getting the Hermitage bonus from your religion, which makes that a wasted follower belief).
I see you built your Musican's Guild quite early, so you have more Great Works of Music than you want or need at this stage of the game, and it means naturally spawning Great Musicians will come much more slowly -- you need them to perform concert tours with the Mayans. You have completed Aesthetics, so you are able to buy Great Musicians later in the game to perform the concert tours. But if you want to buy more than one Great Musician you will need to ramp up your faith output. You only need two more temples (plus one shrine) to allow you to build the Grand Temple -- those alone will give you an additional 13 faith per turn, and you need it (you're currently only generating 16 faith). There's only one religious CS on the map and they are both hostile and in Mayan's back pocket, so I wouldn't waste gold on them (although friendship would net you some additional faith).
Finally, in a couple of turns, your World Ideology proposal will go down in flames and International Games will pass. You really need to shift all production to IG and try to win that. And wake up your workers to build railroads; you need the production bonus ASAP.
per tile? What free gold? Please enlighten again.
