You can certainly bring a motion before the court to delay the surrender date by a certain amount of days but you would have to have a compelling reason, provide a date certain as to when you would turn yourself in.
Wanting an extension so you can file an appeal is not a valid reason -- particularly where you already pleaded guilty. There is never a good time to surrender oneself for custody and even the strongest of marriages is adversely affected by that. I do not believe that will be compelling either. Likewise, the fact that you have to support your familiy financially is no basis. From the jude's point of view, you should have considered these things before you engaged in criminality (or, at least, pleaded guilty).
Your best bet, from what you told me, is that their are health concerns with your mother. If you reasonably expect to delay surrender, you should obtain a letter from a doctor which discusses her mental state and how surrendering right now would push her to a breaking point. However, this will likely only be successful if you provide some care for your mother and her condition is transient. (In other words, if she lives you with you, you have a good chance; but if she lives 500 miles away, not so much. Additionally, if she is under treatment and will be better in 90 days, you have a decent shot at delay; but if there will never be a good time for you to surrender, there is no reason to delay it now.)
Finally, do not indicate to the judge that you pleaded guilty to something that you did not do. Even if true, this will be seen as a failure to accept responsibility and turn the judge off to your situation.