I mean the theory has been around for over a century now and the experiments conducted are real. It hasn't been studied in-depth but you can't really write off the results as fake. Whether there is another possible explanation is yet to be seen but nobody has proposed an alternative AFAIK.
BTW found the study on flies. It's from more than half a decade ago but
here you go for anyone that's interested.
In case anyone is interested in that study here's as concise an explainer as I could manage:
The "setup": They mated female flies with two males (smaller and larger ones) at separate points in time. At first when the female wasn't fertile yet and then two weeks later as the eggs fully developed, covering all four scenarios (first the larger male and then the smaller one, as well as vice-versa and without changing groups)
Their hypothesis: Sperm has an effect on eggs that aren't yet ready to be fertilized by sperm, by depositing in the eggs (see attached image)
Their conclusion: They express the hypothesis is credibly supported by their findings, because an increase in body size was observed even if the smaller male is genetically the father.
Important: The offspring does not share genetic markers with both fathers. In both scenarios the vast majority of offspring are sired by the male who mated with the female later, because sperm can potentially remain alive unexpectedly long/eggs sometimes develop unusually early some of the "early birds" passed on their genes too, but never both to a single individual.
Some problems I encountered (and to their credit the authors acknowledge some of this too): It is not irrefutably proven that this is the exact causal mechanism, to put forth a theory that (unproven as it is) would still be supported by their data: The female absorbs the sperm and uses the nutrients to kick-start the offsprings development (again, I made that up), but their findings would be equally indicative of that hypothesis being true. Also age is a significant factor in how sperm achieves fertilization (the older the less potent generally) which is something that matters significantly for something as shortlived as a fruitfly, which might have introduced further noise into the data. At best further studies are necessary. Something I noticed as well is their focus on size, which doesn't make much sense to be the only factor influenced by the first (attempted) mating process. If sperm can influence development in eggs that are fertilized by different sperm at a later point, why would it only affect size and not, say, colour?
Additionally: Insect and mammal biology differs quite drastically (duh), their hypothesis might be true for fruit flies and still not apply to human reproduction. But even if that where the case: Remember women menstruate, your girl would have to cheat on you during or in the few days after her period and then get impregnated by you in the same month for this to work in the same way as is outlined in the study. Supposing all that would work (doubtful, to put it mildly): Your children would still not share genetic code with the other guy.
My personal conclusion: Chose something else to worry about.
