The ý is wrong here, but the acute accent prolongs the vowel sound for half again its usual value.
The s with the hook sign "háček" [š] above it gives it the phonetic value "sh".
Likewise the č: its phonetic value is "ch" as in "choose".
There should be no acute accent over the e in the cat's name.
Almost all Czech sentences have the stress on the first syllable. There is a difference in Czech between stress and prolongation of the phonetic value. A long a [á] is likely not stressed, but it has got to be prolonged in pronunciation. This is often difficult for non-Czech speakers to differentiate and master. The letter "y" and the letter "i" both are sounded as in Italian:hard e. There are subtle differences between these two that Czechs can hear. Likewise Czechs can hear the difference between y and ý.
The cat's name is in the DIMINUTIVE. In English we have Richard and Dick and Dickie. Czech diminutives evince affection and are vary common. So here we have formed the cat's name from the Czech word for plush, "plyš", and appended the affectionate diminutive; the cat's name is thus "little plush or plushie".
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