Send Ajax Array to Controller

You could start by defining a view model on the server that will represent the data you would like to retrieve:

public class MyViewModel
{
    public Guid EmployeeId { get; set; }
    public int Year { get; set; }
    public int Month { get; set; }

    public ItemViewModel[] Items { get; set; }
}

public class ItemViewModel
{
    public int TotalHours { get; set; }
    public int TotalMinutes { get; set; }
}

and then have your controller action take this view model as argument:

[HttpPost]
public ActionResult UpdateEmployeeClockedHoursByProvider(MyViewModel model)
{
    ...
}

The next step is to fix your view a little bit because right now you seem to be hardcoding input fields instead of using html helpers and defining wrong ids and names for your input fields (in HTML the id and name attribute cannot start with a number).

So here's how you could generate the table:

<% using (Html.BeginForm("UpdateEmployeeClockedHoursByProvider", null, FormMethod.Post, new { id = "myForm" })) { %>
    <table>
        <thead>
            <tr>
                <th>Clocked Date</th>
                <th>Total Hours</th>
                <th>Total Minutes</th>
            <tr>
        </thead>
        <tbody>
            <% for (var i = 0; i < Model.Count; i++) { %>
            <tr>
                <td>
                    <%= Html.DisplayFor(x => x[i].ClockedDate) %>
                </td>
                <td>
                    <%= Html.TextBoxFor(
                        x => x[i].TotalHours, 
                        new { 
                            type = "number", 
                            placeholder = "Hours", 
                            @class = "total-hours" 
                        }
                    ) %>
                </td>
                <td>
                    <%= Html.TextBoxFor(
                        x => x[i].TotalMinutes, 
                        new { 
                            type = "number", 
                            placeholder = "Minutes", 
                            @class = "total-minutes" 
                        }
                    ) %>
                </td>                    
            </tr>
        <% } %>
        </tbody>
    </table>

    <button type="submit">Save</button>
<% } %>

and finally you could have some javascript file that will subscribe to the .submit handler of the form and send the values to the server:

$(document).on('#myForm', 'submit', function () {
    var items = [];
    $('table tbody tr').each(function() {
        items.push({
            totalHours: $(this).find('td input.total-hours').val(),
            totalMinutes: $(this).find('td input.total-minutes').val()
        });
    });

    var data = {
        employeeId: employeeId, // <!-- It's not very clear from your code where is this supposed to come from
        year: $('#year').val(),
        month: $('#month').val(),
        items: items
    };

    $.ajax({
        url: this.action,
        type: this.method,
        contentType: 'application/json',
        data: JSON.stringify(data),
        success: function (result) {
            // TODO: here you could handle the response from the server
            alert('Your request has been successfully processed.');
        },
        error: function (xhr, ajaxOptions, thrownError) {
            alert(xhr.status);
            alert(thrownError);
        }
    });

    return false;
});

In this example notice how I used the .on() method instead of .live() which is deprecated and should not be used anymore.