IIRC the Greens defence spokesperson pressed for either no weapons, or no standoff weapons, to be acquired for the P-8A Poseidons. At present, AFAIK no aAShM apart from the Penguin is in the NZDF inventory, nor are there any plans for this to be changed anytime soon. Also, AFAIK (and happy to be corrected if I have this wrong) the Penguin has not been fitted or integrated with the Poseidon and there are no plans to do so given the age of the Penguin and the size of its user base. IIRC the later Mk 3 (RNZAF has ex-Oz Mk 2 Mod 7) Penguins have a range of ~60 km when fired from a helicopter at altitude. That sort of missile launch range would put any launching aircraft well within the range of long-range ship mounted area air defences, and potentially within the range of medium-to-long range missile defences as well. In short, any naval vessel with a role like that of the RAN's Hobart-class DDG would be able to engage a potential launch aircraft before the aircraft could get within launch range to use any Penguin AShM. So yes, whilst the NZDF might have an AShM in inventory, it really is not one which can be relied upon for use. There is a reason why Kongsberg (maker of the Penguin) developed the NSM which entered into service as a Penguin replacement, 40 years after the Penguin entered service. The NSM is a slightly heavier missile, somewhere between 25 kg and 40 kg more than the Penguin, depending on Penguin version, but has a standoff range of 100+ n miles.
My take, given how often the NZDF seems to have problems getting new/replacement ordnance, is that there are ideological objections, which then cause funding and/or priority to be denied to needed ordnance programmes. I recall not long after I first joined, reading through the then LTDP which noted that NZDF's stock of Mk 46 LWT's at the time were set to expire in the near future, IIRC on or before 2010, but the LTDP did not call for replacing the LWT stocks until 2015 or thereabouts. In effect, the official plan at the time was to continue (to attempt) fielding expired ordnance for several years, whilst ignoring the potential risks associated with carrying expired ordnance. As I understand it, the risks were that if the NZDF needed to use an expired Mk 46 (as opposed to one that was inspected/re-manufactured to ensure that it was good) there would be the potential that a launched Mk 46 would be a dud.
Given the current lack of known plans to either arm the P-8 Poseidons set to enter service, or be mounted to ships and then queued from the Poseidons... It seems that the NZDF will have a great tool for maritime domain awareness, but still have little ability for any sort of response.