Let's say I play a game or sport against somebody stronger than me. I might plausibly feel a need to raise my game. Are there any empirical academic studies which test whether such a phenomenon exists?
According to Tenenbaum et al. (1995) the effect may be opposite. In their study, they had people, who where normally not engaged in a training program for strength, tested on their knee-extensor strength before and after four sessions of psychological interventions.
There were three conditions. In the first, the control group, no intervention was given. In the second condition, people received positive statements. The third group of people received "relaxation-visualization autogenetic training".
The surprising result were that the control improved significantly in "peak force", and the improvement was significantly greater than the intervention groups. No differences in increment where found on "peak power".
In short, the authors found that mental preparation or "psyching up" is not beneficial for physical performance. The authors argued that "psyching up" may distract you from the task you should be performing, and that that may have a negative influence on you performance. Do take into account that the study is very old and that the result may be outdated. However, the paper may be a nice start for further reading into this topic.
Tenenbaum, G., Bar-Eli, M., Hoffman, J. R., Jablonovski, R., Sade, S., & Shitrit, D. (1995). The Effect of Cognitive and Somatic Psyching-up Techniques on Isokinetic Leg Strength Performance. The Journal of Strength & Conditioning Research, 9(1), 3-7.