Kim's father and namesake Kimball O'Hara dies near the beginning of the book, and he certainly doesn't appear to have much of an influence on Kim himself (that we can see, at least). He's a former member of the Irish Mavericks regiment, and he was also a worker on the Ferozepore (now Firozpur) railway.
But by the time the book starts, he is an opium addict given to hallucinations. In his long rants, he recommends that Kim keep an eye out for a Red Bull on a green field, but he is too incoherent to explain what this Red Bull actually means. It is only in Chapter Five that we find out that Kimball O'Hara is talking about the flag of his old regiment.
We do get the sense, since Kim takes his father's supposed prophecy so seriously, that there is love between them. But by the time Kim sets out on the road with the lama, he does not seem to think of his father regularly at all. In fact, it's partly because Kim is an orphan that he has the freedom to pursue his destiny as the Perfect Spy.