Gonococcal Urethritis: Although the patient has a white discharge, which is more commonly associated with non-gonococcal urethritis, the history of unprotected sex increases the likelihood of a sexually transmitted infection (STI) such as gonorrhea. The presence of pus cells in the urine supports this diagnosis, despite the absence of symptoms like urethritis or ulcers.

Non-Gonococcal Urethritis (NGU): This condition, often caused by Chlamydia trachomatis or Mycoplasma genitalium, is a common cause of urethral discharge in men, especially after unprotected sex. The absence of nitrite in the urine makes a bacterial infection like gonorrhea less likely, but NGU can still present with pus cells.
Trichomoniasis: Caused by the protozoan Trichomonas vaginalis, this STI can cause urethral discharge in men, although it's more commonly associated with vaginal discharge in women. The white discharge and history of unprotected sex make this a plausible diagnosis.

Chlamydial Urethritis with Complications: Although less likely given the absence of symptoms like urethritis, missing a diagnosis of chlamydial infection could lead to complications such as epididymitis or reactive arthritis if not treated promptly.
Syphilis: In its primary stage, syphilis can cause a painless ulcer (chancre) but might not always be visible or recognized. A white discharge is not typical, but syphilis is a critical diagnosis not to miss due to its potential for severe long-term complications.

Urethral Candidiasis: More commonly seen in diabetic or immunocompromised patients, but could be considered in the differential due to the white discharge, although the patient's profile makes this less likely.
Mycoplasma genitalium Infection: While not as rare as some other causes, this bacterium is increasingly recognized as a cause of NGU. It's included here because it might not be the first consideration but should be kept in mind, especially with the advent of more sensitive diagnostic tests.