You could parse it this way:
全力ダッシュでこっちにやってきた is a relative clause that modifies メガネの男.
= a man with glasses (who dashed here)
全力ダッシュでこっちにやってきたメガネの男が放つ is another relative clause that modifies 唾.
= the saliva (which the man with glasses who dashed ...
I think a better translation for this might be:
The saliva released by the guy with glasses dashing over here (as fast as he could) made me move my bentou out of the way.
That is, it wasn't the guy making (forcing) you to move the bentou, it was the presence of the saliva which caused you to choose to move it, hence the causative of する.
You can take this 「せる」 as the causative or imperative form, but the context determines which. It is a bit complicated.
According to コトバンク:
「息子を公立学校に行かせてください」 : "Please make your child go to a public school"
Your son might want to go to a private school for some reason. In this case, you can't allow him to go ...