TY Pup is a well-known bright eclipsing binary with an orbital period of 0.8192 days. New light curves in B, V, (RI)(C) bands were obtained with the 0.61 m reflector robotic telescope (PROMPT-8) at CTIO in Chile during 2015 and 2017. By analyzing those photometric data with the W-D method, it is found that TY Pup is a low-mass-ratio (q similar to 0.184) and deep-contact binary with a high fill-out factor (84.3%). An investigation of all available times of minimum light including three new ones obtained with the 60 cm and the 1.0 m telescopes at Yunnan Observatories in China indicates that the period change of TY Pup is complex. An upward parabolic variation in the O - C diagram is detected to be superimposed on a cyclic oscillation. The upward parabolic change reveals a long-term continuous increase in the orbital period at a rate of dP/dt = 5.57(+/- 0.08) x 10(-8) days yr(-1). The period increase can be explained by mass transfer from the less massive component (M-2 similar to 0.3 M-circle dot) to the more massive one (M-1 similar to 1.65 M-circle dot). The binary will be merging when it meets the criterion that the orbital angular momentum is less than three times the total spin angular momentum, i.e., J(orb) < 3J(rot). This suggests that the system will finally merge into a rapid-rotating single star and may produce a luminous red nova. The cyclic oscillation in the O - C diagram can be interpreted by the light-travel time effect via the presence of a third body.