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Asgore comes across as a goof, a complete wreck of a man who gives away his flowers instead of selling them, to the point where the Mayor Carol is apparently threatening to kick him out. His constant attempts to woo his former wife Toriel are as doomed as they are cringeworthy. He comes across as an all-around loser who can't get his act together, neither for love nor for money, who has to resort to eating free pickles from Sans, the man who we now know is cavorting with Toriel.

...and yet.

Asgore was the chief of police, before he was apparently forced to step down in disgrace (the details of which are unclear to us due to Kris's avoidance of the topic) - this is a role that would require a great deal of competence and expertise, not to mention strength and dedication. It's mentioned that Undyne, his successor, struggles to fill his boots in the position, which suggests that he was respected in his role, and supposedly did a satisfactory job in it.

Now, in Chapter 4, we see Asgore emerge from the bushes after church to try and gift Toriel yet another bouquet. She, of course, beats a hasty retreat, whereupon he gives Kris a single rose (remember this - this will be important later).

Then, at the Holiday Residence, while clambering through the air vents as the SOUL, we can find Asgore in the bathroom, apparently employed by Carol as a housekeeper, at which point he'll talk about settling something "once and for all" and that he's "got to look again". We can find him again in Carol's bedroom, mentioning something about showing Toriel "what really happened" and that she'll "have to believe me" that he was "just trying to protect everyone."

It's clear that Asgore is using his access to the Holiday household in order to find some sort of evidence that will shed light on an unspecified event... which we would assume to be Dess's disappearance, the event that would have seemingly resulted in his dismissal from the force. All of this to say that Asgore is not quite the complete failure of a person he's made out to be in prior chapters - he's looking for something, and he has a plan to set some kind of record straight and make it so that his family can be happy again.

Right, and what does any of this have to do with Kris, exactly? Why is the giving of a single rose to his child significant? Well, the short answer is - I think that Asgore knows, or has figured out, that Kris is being manipulated and coerced by Carol. It's not much of a stretch to assume this - Kris is his child who he loves dearly, enough to bearhug them when we first meet him in Chapter 1, and he could have deduced that something was amiss with them through his skills as a former police chief. But he wouldn't be able to act directly to help them, seeing how Carol has such a tight leash on the both of them. So they'd have to communicate in secret.

In code.

Look again at the screenshot at the top of this post. Look at how Asgore is winking, and how he says "Our secret." We didn't think about it before, because we just assumed Asgore was being his usual hopelessly Divorced self... but stop now and consider how ridiculous this premise is. It's NO secret whatsoever that Asgore gives his flowers away, and many of them to Toriel. On top of that, there's a decent chance he KNOWS she won't even look at them, even if Kris DID give them to her. Because they were never meant for Toriel in the first place.

Asgore is giving flowers to Kris as a coded form of communication - something that I expect he'd have good knowledge of as, again, a former chief of police. And it's not unreasonable, again, to assume that perhaps Kris might have picked up on some of this, growing up with their adoptive father being in the force. And Carol would have great difficulty detecting the code in the first place, because that's just what bumbling old Asgore does - he gives flowers away to EVERYBODY. And even if she thought something was up, surely he's not sending coded messages to everyone in Hometown, right? Of course not - they're chaff, noise to obscure the fact that he is communicating with Kris right under her nose.

We can skip the bouquet in chapter 1, but we cannot skip the rose in chapter 4. Whatever message he meant to impart to Kris, they WILL have picked up on it, and it's something that may become significant in Chapter 5, which is heavily implied to take place in Asgore's flower shop.