Yes, but it won't do much
Sustain rules say:
If the spell’s duration is “sustained,” it lasts until the end of your next turn unless you use a Sustain a Spell action on that turn to extend the duration of that spell.
Clinging Ice has a duration of Duration sustained up to 1 minute
and says:
Once this spell ends, the target becomes temporarily immune for 1 minute.
If you don't sustain the spell, it will last until the end of your next turn. During that next turn, it has not ended yet, so the target is not immune yet.
However, there is the general rule on Duplicate Effects:
Duplicate Effects
When you’re affected by the same thing multiple times, only one instance applies, using the higher level of the effects, or the newer effect if the two are the same level. For example, if you were using mage armor and then cast it again, you’d still benefit from only one casting of that spell. Casting a spell again on the same target might get you a better duration or effect if it were cast at a higher level the second time, but otherwise doing so gives you no advantage.
So, yes the target is not immune, but your second casting will still do nothing beyond dealing damage, as only one instance of the spell applies. Since they still become immune after the first intance ends at the end of the turn, you won't get the sustain effects of the second casting, and no more damage after that for the next minute:
If you have immunity to a specific condition or type of effect, you can't be affected by that condition or any effect of that type.
