create a new Integer object that holds the value 1?

You skipped the intended solution:

Integer p = Integer.valueOf(1);

This pattern is known as Factory method pattern. One may ask what the benefit of this method is. Luckily, the implementation of class Integer is open-source, so let's take a look:

    public static Integer valueOf(int i) {
        if (i >= IntegerCache.low && i <= IntegerCache.high)
            return IntegerCache.cache[i + (-IntegerCache.low)];
        return new Integer(i);
    }

There seems to be some kind of Integer-value cache. If one requests an Integer with a value within the cache-range, Java does not create a new object, but returns a previously created one. This works because Integers are immutable. One can even control the upper cache limit with the system property java.lang.Integer.IntegerCache.high=....

And why do the other two methods of creating an Integer generate a warning? Because they were set deprecated with Java 9.

Integer#Integer(int value):

Deprecated. It is rarely appropriate to use this constructor. The static factory valueOf(int) is generally a better choice, as it is likely to yield significantly better space and time performance. [...]

Integer#Integer(String s):

Deprecated. It is rarely appropriate to use this constructor. Use parseInt(String) to convert a string to a int primitive, or use valueOf(String) to convert a string to an Integer object. [...]

And just for completeness, here is the part for Integer.valueOf(int i):

Returns an Integer instance representing the specified int value. If a new Integer instance is not required, this method should generally be used in preference to the constructor Integer(int), as this method is likely to yield significantly better space and time performance by caching frequently requested values. This method will always cache values in the range -128 to 127, inclusive, and may cache other values outside of this range.

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EDIT 1: Thanks to @VGR mentioning that

Integer p = 1;

is equivilant to

Integer p = Integer.valueOf(1);

This, however, is only true for int-values between -128 and 127. The behaviour is defined in JLS §5.1.7:

[...] If the value p being boxed is the result of evaluating a constant expression (§15.28) of type boolean, char, short, int, or long, and the result is true, false, a character in the range '\u0000' to '\u007f' inclusive, or an integer in the range -128 to 127 inclusive, then let a and b be the results of any two boxing conversions of p. It is always the case that a == b.

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EDIT 2: Thanks to @DorianGray, who brought the following to my attention.

While not in the JLS, the version of javac I am using (9.0.4) does compile the boxing down to Integer.valueOf(...); as it is shown in this answer by Adam Rosenfield.

Method 4, Integer p = Integer.valueOf(1); is the recommended way. The JavaDoc says:

Returns an Integer instance representing the specified int value. If a new Integer instance is not required, this method should generally be used in preference to the constructor Integer(int), as this method is likely to yield significantly better space and time performance by caching frequently requested values. This method will always cache values in the range -128 to 127, inclusive, and may cache other values outside of this range.