I have a big problem with partial pressure in liquids. I went to wikipedia and I find out that there were no such thing, only partial pressure in a mixture of gas above a liquid and concentration of disolved gas in a liquid (linked by Henry's law).
Correct.
Nevertheless, in many articles I work on, they talk about partial pressure in water and blood, which I don't understand.
Neither do I. Most likely it is part of a jargon used in the branch and not necessarily following the general meaning of the term. Nothing unusual in some applications, but confusing for newcomers.
This is tricky, as I can think of two slightly different meanings. One, is "same as the partial pressure of the gas above the liquid", second is "calculated assuming external pressure and solubilities of the gases" (that is, you treat all the dissolved gases together, calculate their total amount, calculate molar fraction of the oxygen, and then multiply it by the external pressure). These are not equivalent.
I would ask the supervisor.