Step 1: This is an exception question, so the task is to find the one disease the sandfly does not carry.
Step 2: Sandflies of the genus Phlebotomus are classic vectors of leishmaniasis in all its forms: visceral leishmaniasis or kala-azar through P. argentipes, and cutaneous leishmaniasis or oriental sore through P. papatasi and P. sergenti.
Step 3: They additionally transmit the viral sandfly fever. Thus three of the four options - oriental sore, leishmaniasis and kala-azar - are genuine sandfly-borne conditions.
Step 4: Relapsing fever, caused by Borrelia, is carried by lice (louse-borne type) or soft ticks (tick-borne type), never by the sandfly, making it the odd one out.
\[\boxed{\text{Relapsing fever}}\]