As an ordinary Chinese who was born, grew up, educated, and worked in China, I think the state of free speech in China is far worse than one who is living in a world with freedom of expression could imagine.
This isn't just my personal perception. From the Democracy Index provided by The Economist, we can see that China's ranking has been declining over the past decade.
Can they, for instance, control the media to publish certain news articles?
Yes, not only can they control what the media reports, but also control what it doesn't. They can control everything.
If so, how do they do so?
For an individual, they can imprison you for a variety of reasons, such as solicitation. Moreover, China has a notoriously vague law known as "picking quarrels and provoking troubles".
Apart from censorship is there any tool in their toolbox that the government can use to control the narrative and manipulate the public opinion
Internet water army, also known as 水军,网评员,五毛. When news unfavourable to the CCP arises, and it's not convenient to remove it for various reasons (such as its significant impact), the CCP employs online trolls to steer public opinion. A common tactic is to glorify hardship.
For instance, if a delivery driver is overworked due to the government's inaction and lack of legally mandated labor protections, Internet water army would praise his diligence, diverting attention away from the government's failure to provide adequate labor safeguards.
I am wondering if they can also control other organizations in the media industry and how.
By exerting pressure on businesses.
If you run a business in China, making a profit while strictly adhering to the tax laws is almost impossible. You'd likely need to break the law in some aspects to make money. This legal framework is intentionally designed by the CCP to ensure they have leverage over every enterprise.
Often, when your posts are removed from Chinese social media platforms, it's an action taken by the platform itself, not a directive from the Communist Party. These platforms are merely trying to protect themselves.
Some platforms, which have been penalized multiple times by the Communist Party, even require individual review of comments before they become visible to others. eg: IT之家
As an ordinary person living under an authoritarian regime, the genuine thoughts and feelings of the Chinese people are seldom understood by foreigners. I've never expected individuals from the free world to be devoid of prejudice towards me; I simply wish that when you look at China, you distinguish between its people and the Chinese Communist Party.
Human nature is universal. It's for this reason that the Chinese Communist Party fears freedom of speech just as the German Nazis, and the Soviet Communists did. For the same reason, the Chinese people also yearn for freedom and democracy, much like the people of East Germany and Romania once did.
