Two terms to define here first:
Wild-animal suffering is the idea that animals in the wild experience some amount of suffering naturally, e.g. from parasites, exposure, hunger, being killed slowly by predators, etc. Some argue that the life of an average wild-animal (especially when you consider marine animals and insects) is so full of suffering that they experience more suffering than wellbeing. This might lead to the conclusion that their lives are not worth living, and would be better off not being born, so to speak. (Note this doesn’t automatically mean we should kill all predator animals, as some strawman makers of this would argue)
Speciesism I’ll leave to Peter Singer to define (from his book Animal Liberation): “a prejudice or attitude of bias in favor of the interests of members of one’s own species and against those of members of other species”. It is a similar idea to racism, sexism, or any other ‘ism’.
Many argue (and I’d agree) that causing harm to animals for small amounts of human pleasure (such as eating their flesh or secretions) is speciesist. I prefer the utilitarian framework, but I concede that this is speciesist as much as the mistreatment of other races would be racist.
I’ve seen recently some people argue that thinking we have the right to intervene in the lives of wild animals in any way to try and alleviate suffering is speciesist. I argue here the opposite.
When a human is intentionally harmed by another human, we naturally think that this is bad. Most people also believe that a human intentionally harming a non-human is bad (though some will exempt certain animals from this care!). When a human suffers through some natural cause, e.g. exposure, hunger, disease, we tend to also think this is bad, and will do our best to help them. Why should we think that the same suffering, experienced by a wild animal, is not bad, or that we shouldn’t also try to prevent it?
Suffering is bad regardless of the cause, as the individual experiencing the suffering doesn’t intrinsically care where the suffering came from. And so I argue that caring about natural human suffering but not natural non-human suffering is speciesist.