Are you asking whether it's OK to miss off the final 's' from the word 'asks'? If so, the answer is most definitely 'no'.
Native speakers never omit the 's' at the end of words, however quickly we're speaking. As native speakers of English, we've grown up pronouncing consonant clusters such a 'sks', and they don't feel 'unnatural' to us. Nor is there any need to 'disrupt' the sentence by pronouncing this group of consonants, as the 't' of 'Tom' flows quite naturally on from the 's' of 'asked'. If you're breaking the flow of the sentence to make these sounds you are probably over-enunciating them. And in the case of the past tense, the 't' sound at the end of 'asked' is merged into the 't' of 'Tom', so there's no problem there, either.
In fact, if any sound is dropped in connected speech, it's the 'k' in 'asked' or 'asks'. It's not uncommon for a native British English speaker, for example, to pronounce 'asked' as /ɑːst/ when speaking in a relaxed manner. So, the second and third words of 'She asked Tom to go on a date' might run together as /ɑːstɒm/ .