TL;DR: actions that society considers morally reprehensible and "corrupt" when carried out by public institutions, are seen as normal and acceptable when it comes to private institutions, so traditional comparisons of "corruption" between capitalism and socialism put socialism at a disadvantage by definition.
1- Let us imagine that I'm a business owner, and I decide to carry out some renovations in my building. I decide that, since I have a reliable friend who owns a renovations company, I will simply carry out the renovation with their company. We sign a contract, the renovation is carried out, the work gets done, the other company gets paid. Nothing out of the norm here.
Now let us imagine the case in which this first business, instead of being owned by me privately, is socialized and owned by the state: a public entity. Some renovations are necessary, so I, as a public administrator, decide to order the renovation to be carried out by a friend... except that's corruption! I need to organize an auction and order impartially from a variety of firms, by lowest expense and by highest level of satisfaction! What is normal and approved in capitalism, is unthinkable and in most instances illegal under the principles of public ownership!
2- Another example: I'm a worker in a private company. One year, the CEO that is put in place by the stockholders, happens to be a former employer of mine, and because they know me and my performance, I get promoted. Meritocracy! Some people even call that "networking", which is a necessary social skill in capitalism and highly regarded in wealthy circles.
Now let us imagine the case of a soviet workplace, in which I'm a worker with excellent performance. The union-approved party member in charge at this time, sees my performance and my contributions and involvement with the union and party, and decides to offer me a promotion. Oh, what a blatant case of dictatorial bureaucracies, in which only party members giving each other favours get to rise to the top! What an unfair and corrupt system!
Whenever we hear these claims of "corruption", "bureaucracy" and such from socialism, please make it a point to compare these events with similar instances in capitalism, and how normalized and approved by the social majority they are. Why do we only expect transparency, efficiency and impartiality from public institutions, but normalize the opposite behaviours in capitalist enterprises?
But private corporations do have trust placed in them, and have a responsibility to the public. My own country recently privatized our energy company, and sold it to an incompetent private company. That private company is constantly making decisions to minimize costs, which results in poorer service and constant blackouts. But it's 100% legal for them to be incompetent and corrupt, while a public energy company would be held to a higher standard.
The logic here is that if a private firm is corrupt, it'll be less efficient than a firm that isn't corrupt, and is marginally outcompeted. A public entity has a monopoly by nature and cannot be outcompeted, so corruption in a public entity is inherently more problematic. But an elementary tendency that Marxists have been shouting about for centuries is that there is no free market competition under capitalism in the long term: in a competition, someone wins, and captures the market share of all the competitirs. In practice, a private corporation will eventually have the same monopoly that a public entity has, with none of the legal responsibility.